A message from Sal Khan for the Khan Academy 2016 Annual Report

WEBVTT
Kind: captions
Language: en

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- [Instructor] Welcome to the Khan Academy

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2016 Annual Report.

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In the actual text of the report,

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we're going to go into a lot more detail

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on the financials and other things

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but I'm hoping here to
give you an overview.

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Big picture 2016 was a
great year for Khan Academy.

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The impact has continued to grow

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in the double digit percentages
and is accelerating.

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Over a 100 million people
used our platform last year,

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people from almost every
country in the world

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and they spent an aggregate
of 5.6 billion total minutes.

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So one of the areas that
we've really started

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to accelerate in 2016 was
in content development.

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Many people associate Khan
Academy with mathematics

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but what we've really been investing in

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is going beyond mathematics.

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We really wanna cover all of
the core academic subjects

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that you consider to be
part of your education

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all the way through the core of college

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and it's not just videos.

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We're doing a lot on practice

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and we're doing a lot of written articles.

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2016 was a very big deal

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in terms of a partnership
with the College Board.

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A few years ago, they
reached out to us, said,

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we are going to revamp the SAT

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and as part of that, we want to address

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at least the perception of inequity

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that's been around for decades

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and the reason why they wanted
to partner with Khan Academy

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is traditional test prep was
associated with gaming an exam.

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It wasn't about real learning

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but they saw that we were
focused on filling in gaps

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in students' understanding.

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So we launched with the College Board

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and what we've seen over
the course of the year

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is Khan Academy is now used more

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than all of paid test prep combined

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and we're seeing that usage
very broad and very deep

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and what's been even more exciting

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is the PSAT which is taken
in 10th or 11th grade

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by 80% of American high school students,

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in the past, that was just a
random practice test you took

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but now it acts as a
diagnostic for Khan Academy.

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You can take your PSAT scores,

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give permission to Khan Academy

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and then Khan Academy's software

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knows your strengths and weaknesses,

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not just in math but in
reading and writing as well

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and can immediately
give you practice items

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that are essentially weak point training

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and even more powerful,

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we've been able to follow
students from that PSAT

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through their work on Khan Academy

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to eventually taking the SAT

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to understand the efficacy
of this entire ecosystem

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and what you see here on the slide

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is some of the preliminary
results for arguably the largest

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personalized learning
study ever conducted.

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With over 300,000 students,

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we saw that those who
prepared on Khan Academy

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for at least 20 hours

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saw twice the growth
from the PSAT to the SAT

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than they would typically see

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and this is just the beginning.

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We're working with the
College Board very closely

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to do even further studies

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and we've also seen that it
doesn't stop at 20 hours.

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The more time that
students put in practice,

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the more gains we're seeing.

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Another really exciting event in 2016

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is that Duck Duck Moose
became part of Khan Academy.

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Duck Duck Moose, I would argue,

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was the best early learning
app company out there.

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In 2016, the founding team

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said we would rather see our
work having a social mission

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and reaching as many students as possible

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especially those that are most in need

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and so they became part
of our team in 2016

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and going forward,

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we're going to be working
on early learning apps

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that holistically approach
reading, writing, mathematics

00:03:26.787 --> 00:03:28.586
starting at pre-kindergarten

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and then taking students
through early elementary school.

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International continues
to be a very big part

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of Khan Academy where
roughly 1/3 of our usage

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is outside of the United States.

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We have increased our focus
on India, Mexico and Brazil

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and going forward, we're
building tools and platforms

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to allow each geography
to be even more localized

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to their specific needs

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and going forward, there's
a lot of exciting things.

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What you see here is a screenshot

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for the student facing experience
for back to school 2017

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where students can tell Khan Academy

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what classes they're taking,
what grade they're in

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and then Khan Academy
can personalize to them

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and also have game mechanics

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so that the students are
motivated to move forward.

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You can see that right over here

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once you get into a unit page.

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Of course, you see the videos,

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you see the practice items here

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but we are also adding diagnostic quizzes

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and unit tests as well.

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What you see here is our
assignments functionality

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where a teacher can assign
any piece of content

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on Khan Academy.

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This is a teacher assigning
some practice content.

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They can assign it to particular students

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in a particular class.

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They can set a due date.

00:04:42.679 --> 00:04:44.026
They can then see a readout

00:04:44.026 --> 00:04:47.776
of which students
completed which assignments

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and then they can see which items

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gave the students the most difficulty

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so that they can review
it in class the next day.

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On top of all of this,

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we have a program called LearnStorm

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which is a learning challenge

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that schools, teachers,
students participate in

00:05:03.975 --> 00:05:06.146
that celebrate the growth mindset.

00:05:06.146 --> 00:05:08.562
The more students can
show grit, perseverance

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as they work through
practice on Khan Academy,

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the more awards and
recognition the students,

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their classrooms, their teachers
and their schools will get.

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We've piloted it in several
geographies in the past

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and 2017 is gonna be the first time

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that we do this on a nationwide scale.

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So there's a lot going on at Khan Academy.

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What I just covered is
really just a sample of it

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but we're really grateful
for your support.

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We have a big mission.

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A free, world-class education
for anyone, anywhere

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and some of y'all have been
with us for many years.

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When we were just starting in
2007 out of a walk-in closet,

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this type of a mission would
have seemed frankly delusional

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but because of your
support, your partnership,

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all of the work that we've
been able to do together

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with governments, with
teachers, with students,

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it's no longer just a dream.

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It's really happening

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and the momentum that we've
built together in 2016

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is continuing to accelerate into 2017

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and I expect it's going to
accelerate in 2018 and beyond

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so that as we look back
in a decade or two,

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we'll say, how could
we ever imagine a world

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where there wasn't a free
world-class education

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for anyone, anywhere?


Horizontal area between curves

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Language: en

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- [Instructor] So I have
two curves graphed here.

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And we're used to seeing things
where y is a function of x,

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but here we have x as a function of y.

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In fact, we can write this top expression

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as being a function of y.

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And this second one, just
to make it different,

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we could view this as g of y.

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Once again, it is a function of y.

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And what we're concerning
ourselves with in this video is

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how do we find this area

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in this light blue color
between these two curves?

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And I encourage you to pause the video

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and try to work through it.

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All right, so a huge hint here is

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we're going to want to
integrate with respect to y,

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a definite integral where
our bounds are in terms of y.

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So for example, this is,

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this lower point of
intersection right over here,

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this would be our lower
bound in terms of y.

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Let's call that y one.

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And then,

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this up here, this would
be our upper y bound.

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So if we think about where do
these two curves intersect,

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and we look at the y-coordinates
of those intersections,

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well, that gives us two nice
bounds for our integral.

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So we're gonna take our
integral from y one to y two,

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from y one

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to y two,

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y two.

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We're going to integrate
with respect to y, dy.

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And so what are we going to sum up?

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Well, when we integrate,
we can think about

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taking the sum of
infinitely thin rectangles.

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And in this case, it would
infinitely flat rectangles

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since we're thinking about dy.

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So dy would be the height
of each of these rectangles.

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And what would be, in this case, the width

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or the length of this
rectangle right over here?

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Well, over this interval,
from y one to y two,

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our blue function, f of y,

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takes on larger x-values than g of y.

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So this length right over
here, this would be f of y,

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f of y,

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this x-value minus this x-value,

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minus g of y.

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So this is going to be f of y

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minus g of y,

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g of y.

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Well, we know what f of y and g of y are.

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Really, the trickiest part is,

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is figuring out these
points of intersection.

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So let's think about where
these two curves intersect.

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They are both equal to x,

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so we can set these two y
expressions equal to each other.

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So we know that negative,

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let me do it in that other color,

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so we know that negative
y squared plus three y

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plus 11 is going to be equal to this,

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is going to be equal to y squared

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plus y

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minus one.

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So let's just subtract all
of this from both sides

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so that on the right side we have a zero

00:02:55.903 --> 00:02:58.298
and on the left side we
just have a quadratic.

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So let's subtract y squared.

00:03:00.452 --> 00:03:01.854
Let's subtract y.

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And then subtract negative
one, which is just adding one.

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And over here, we're going
to do the same thing,

00:03:08.485 --> 00:03:10.151
minus y plus one.

00:03:10.151 --> 00:03:13.100
And what we are left
with is going to be a,

00:03:13.100 --> 00:03:15.153
hopefully, a straightforward quadratic.

00:03:15.153 --> 00:03:19.524
So let's see, this is going
to be negative two y squared

00:03:19.524 --> 00:03:20.441
plus two y.

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Am I doing that right?

00:03:22.628 --> 00:03:24.628
Yep, plus two y, plus 12

00:03:26.328 --> 00:03:27.745
is equal to zero.

00:03:29.095 --> 00:03:32.785
And then this over here I can
factor out a negative two,

00:03:32.785 --> 00:03:36.838
and I get negative two
times y squared minus y

00:03:36.838 --> 00:03:39.049
minus six is equal to zero.

00:03:39.049 --> 00:03:41.084
This we can factor from inspection.

00:03:41.084 --> 00:03:43.245
What two numbers, when we
add, equal negative one?

00:03:43.245 --> 00:03:45.603
When we take their product,
we get negative six?

00:03:45.603 --> 00:03:48.133
Well, that would be
negative three and two.

00:03:48.133 --> 00:03:52.896
So this is going to be negative
two times y minus three

00:03:52.896 --> 00:03:54.313
times y plus two.

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That's just straightforward
factoring a polynomial,

00:03:58.298 --> 00:03:59.233
a quadratic.

00:03:59.233 --> 00:04:00.078
Did I do that right?

00:04:00.078 --> 00:04:02.491
Yep, that looks right, is equal to zero.

00:04:02.491 --> 00:04:04.368
So what are the points of intersection?

00:04:04.368 --> 00:04:07.156
The points of intersection are going to be

00:04:07.156 --> 00:04:11.323
y is equal to three and y
is equal to negative two.

00:04:13.306 --> 00:04:16.642
So this right over here is
y is equal to negative two,

00:04:16.642 --> 00:04:20.642
and then the upper bound
is y is equal to three.

00:04:21.903 --> 00:04:26.248
So now we just have to evaluate
this from negative two,

00:04:26.248 --> 00:04:28.973
all the way until three.

00:04:28.973 --> 00:04:29.806
So let's do that.

00:04:29.806 --> 00:04:30.804
I'm gonna clear this out,

00:04:30.804 --> 00:04:33.612
so I get a little bit of real estate.

00:04:33.612 --> 00:04:37.274
So this is equal to the integral from

00:04:37.274 --> 00:04:38.712
negative two

00:04:38.712 --> 00:04:39.629
to three of

00:04:42.585 --> 00:04:44.168
negative y squared,

00:04:45.384 --> 00:04:47.098
plus three y,

00:04:47.098 --> 00:04:47.931
plus 11,

00:04:49.067 --> 00:04:51.027
minus all of this stuff.

00:04:51.027 --> 00:04:52.804
So if we just distribute
a negative sign here,

00:04:52.804 --> 00:04:54.985
it's minus y squared,

00:04:54.985 --> 00:04:56.485
minus y, plus one,

00:04:57.380 --> 00:04:59.213
and then we have a dy,

00:05:00.123 --> 00:05:01.589
dy.

00:05:01.589 --> 00:05:04.973
This is equal to the
definite integral from

00:05:04.973 --> 00:05:07.806
negative two to positive three of,

00:05:10.677 --> 00:05:13.359
let's see, negative y
squared minus y squared,

00:05:13.359 --> 00:05:15.515
negative two y squared,

00:05:15.515 --> 00:05:20.178
and then three y minus y
is going to be plus two y.

00:05:20.178 --> 00:05:24.395
And then 11 plus one, plus
12, we saw this just now,

00:05:24.395 --> 00:05:27.572
when we were trying to solve for y, dy.

00:05:27.572 --> 00:05:29.890
And so what is that going to be equal to?

00:05:29.890 --> 00:05:32.115
Well, we just take the
antiderivative here.

00:05:32.115 --> 00:05:35.130
This is going to be,
let's see, negative two.

00:05:35.130 --> 00:05:37.226
Let's increment the
exponent, y to the third.

00:05:37.226 --> 00:05:39.579
Divide by that exponent,
reverse power rule,

00:05:39.579 --> 00:05:43.746
plus two y squared divided by
two, which is just y squared,

00:05:44.658 --> 00:05:46.464
just the reverse power rule,

00:05:46.464 --> 00:05:48.047
and then plus 12 y.

00:05:49.876 --> 00:05:53.293
And we're going to evaluate that at three

00:05:54.662 --> 00:05:57.003
and at negative two.

00:05:57.003 --> 00:06:00.862
So if we evaluate that at
three, we are going to get,

00:06:00.862 --> 00:06:03.192
let's see, negative two

00:06:03.192 --> 00:06:04.025
times 27

00:06:05.137 --> 00:06:06.462
over three,

00:06:06.462 --> 00:06:08.072
plus nine,

00:06:08.072 --> 00:06:08.905
plus 36.

00:06:09.813 --> 00:06:14.488
And then we are going to
want to subtract, minus

00:06:14.488 --> 00:06:16.158
all of this evaluated at negative two.

00:06:16.158 --> 00:06:20.325
So it's going to be negative
two times negative eight

00:06:21.654 --> 00:06:22.571
over three,

00:06:24.343 --> 00:06:25.176
plus four,

00:06:26.699 --> 00:06:28.063
minus 24.

00:06:28.063 --> 00:06:30.552
So we just have a little bit
of mathematics ahead of us.

00:06:30.552 --> 00:06:31.443
So let's see,

00:06:31.443 --> 00:06:34.848
this is going to be 27
divided by three is nine.

00:06:34.848 --> 00:06:36.765
So this is negative 18.

00:06:37.849 --> 00:06:38.766
Negative 18

00:06:39.717 --> 00:06:43.329
plus nine is going to be negative nine,

00:06:43.329 --> 00:06:46.999
plus 36, all of that is
going to be equal to,

00:06:46.999 --> 00:06:49.999
so the stuff in blue is equal to 27.

00:06:50.916 --> 00:06:51.896
Right, did I do that right?

00:06:51.896 --> 00:06:55.641
Get negative 18 plus nine, yep, 27.

00:06:55.641 --> 00:06:59.005
And then all the stuff in red over here,

00:06:59.005 --> 00:07:02.098
we have, this is going to be
negative times a negative,

00:07:02.098 --> 00:07:06.115
so it's 16 over three,
plus four, minus 24.

00:07:06.115 --> 00:07:08.198
So that is going to be 16

00:07:09.880 --> 00:07:10.797
over three,

00:07:11.766 --> 00:07:13.016
and then minus,

00:07:14.126 --> 00:07:14.959
minus 20,

00:07:16.487 --> 00:07:18.001
minus 20.

00:07:18.001 --> 00:07:20.446
But then we have this
negative out here, so it's,

00:07:20.446 --> 00:07:22.879
if we distribute that, we'll get plus 20,

00:07:22.879 --> 00:07:25.900
minus, we could say
instead of 16 over three,

00:07:25.900 --> 00:07:28.218
we could rewrite that as 5 1/3,

00:07:28.218 --> 00:07:30.423
minus 5 1/3.

00:07:30.423 --> 00:07:31.964
And so what is that going to get us?

00:07:31.964 --> 00:07:34.558
Let me scroll down a little
bit, or let me go to the right,

00:07:34.558 --> 00:07:37.580
so I have a little bit
more real estate here.

00:07:37.580 --> 00:07:40.838
So then that is going to be equal to,

00:07:40.838 --> 00:07:43.338
get a minor drum roll here, 47

00:07:44.259 --> 00:07:45.092
minus five

00:07:45.964 --> 00:07:47.694
minus 1/3,

00:07:47.694 --> 00:07:51.609
which is equal to, let's see,
47 minus five, that's 42,

00:07:51.609 --> 00:07:55.478
minus 1/3 is, now we get a drum roll,

00:07:55.478 --> 00:07:56.978
41 2/3.

00:08:00.513 --> 00:08:01.846
And we are done.


Area between a curve and and the _-axis

WEBVTT
Kind: captions
Language: en

00:00:00.409 --> 00:00:01.257
- [Instructor] So right over here,

00:00:01.257 --> 00:00:05.582
I have the graph of the function
y is equal to 15 over x,

00:00:05.582 --> 00:00:09.598
or at least I see the part of
it for positive values of x.

00:00:09.598 --> 00:00:12.790
And what I'm curious
about in this video is

00:00:12.790 --> 00:00:15.752
I want to find the area
not between this curve

00:00:15.752 --> 00:00:17.590
and the positive x-axis,

00:00:17.590 --> 00:00:22.131
I want to find the area between
the curve and the y-axis,

00:00:22.131 --> 00:00:26.298
bounded not by two x-values,
but bounded by two y-values,

00:00:27.182 --> 00:00:28.957
so with the bottom bound

00:00:28.957 --> 00:00:32.177
of the horizontal line y is equal to e

00:00:32.177 --> 00:00:36.842
and an upper bound with y is
equal to e to the third power.

00:00:36.842 --> 00:00:39.743
So pause this video, and see
if you can work through it.

00:00:39.743 --> 00:00:41.122
So one way to think about it,

00:00:41.122 --> 00:00:43.960
this is just like definite
integrals we've done

00:00:43.960 --> 00:00:46.441
where we're looking between
the curve and the x-axis,

00:00:46.441 --> 00:00:49.095
but now it looks like
things are swapped around.

00:00:49.095 --> 00:00:52.102
We now care about the y-axis.

00:00:52.102 --> 00:00:55.060
So let's just rewrite our function here,

00:00:55.060 --> 00:00:57.128
and let's rewrite it in terms of x.

00:00:57.128 --> 00:00:59.566
So if y is equal to 15 over x,

00:00:59.566 --> 00:01:02.262
that means if we multiply both sides by x,

00:01:02.262 --> 00:01:03.762
xy is equal to 15.

00:01:04.658 --> 00:01:06.284
And if we divide both sides by y,

00:01:06.284 --> 00:01:09.117
we get x is equal to 15 over y.

00:01:10.108 --> 00:01:14.134
These right over here are
all going to be equivalent.

00:01:14.134 --> 00:01:17.366
Now how does this right over help you?

00:01:17.366 --> 00:01:20.410
Well, think about the area.

00:01:20.410 --> 00:01:21.658
Think about estimating the area

00:01:21.658 --> 00:01:23.193
as a bunch of little rectangles here.

00:01:23.193 --> 00:01:24.709
So that's one rectangle,

00:01:24.709 --> 00:01:28.862
and then another rectangle
right over there,

00:01:28.862 --> 00:01:32.176
and then another rectangle
right over there.

00:01:32.176 --> 00:01:35.034
So what's the area of
each of those rectangles?

00:01:35.034 --> 00:01:38.034
So the width here, that is going to be x,

00:01:38.034 --> 00:01:41.426
but we can express x as a function of y.

00:01:41.426 --> 00:01:43.460
So that's the width right over there,

00:01:43.460 --> 00:01:47.334
and we know that that's
going to be 15 over y.

00:01:47.334 --> 00:01:49.132
And then what's the height gonna be?

00:01:49.132 --> 00:01:52.206
Well, that's going to be
a very small change in y.

00:01:52.206 --> 00:01:54.502
The height is going to be dy.

00:01:54.502 --> 00:01:57.126
So the area of one of
those little rectangles

00:01:57.126 --> 00:02:00.254
right over there, say the area
of that one right over there,

00:02:00.254 --> 00:02:02.354
you could view as,

00:02:02.354 --> 00:02:05.771
let me do it over here, as 15 over y, dy.

00:02:07.972 --> 00:02:10.538
And then we want to sum all
of these little rectangles

00:02:10.538 --> 00:02:12.023
from y is equal to e,

00:02:12.023 --> 00:02:15.380
all the way to y is equal
to e to the third power.

00:02:15.380 --> 00:02:17.341
So that's what our definite integral does.

00:02:17.341 --> 00:02:19.084
We go from y is equal to e

00:02:19.084 --> 00:02:21.608
to y is equal to e to the third power.

00:02:21.608 --> 00:02:24.240
So all we did, we're used
to seeing things like this,

00:02:24.240 --> 00:02:26.808
where this would be 15 over x, dx.

00:02:26.808 --> 00:02:30.976
All we're doing here is,
this is 15 over y, dy.

00:02:30.976 --> 00:02:32.696
So let's evaluate this.

00:02:32.696 --> 00:02:35.777
So we take the antiderivative of 15 over y

00:02:35.777 --> 00:02:37.354
and then evaluate at these two points.

00:02:37.354 --> 00:02:39.046
So this is going to be equal to

00:02:39.046 --> 00:02:42.016
antiderivative of one over y

00:02:42.016 --> 00:02:45.834
is going to be the natural log
of the absolute value of y.

00:02:45.834 --> 00:02:49.254
So it's 15 times the natural log

00:02:49.254 --> 00:02:51.145
of the absolute value of y,

00:02:51.145 --> 00:02:54.303
and then we're going to
evaluate that at our endpoints.

00:02:54.303 --> 00:02:57.400
So we're going to evaluate it at

00:02:57.400 --> 00:02:59.268
e to the third

00:02:59.268 --> 00:03:00.101
and at e.

00:03:01.506 --> 00:03:03.507
So let's first evaluate at e to the third.

00:03:03.507 --> 00:03:06.749
So that's 15 times the natural log,

00:03:06.749 --> 00:03:09.891
the absolute time, the natural,
(laughs) the natural log

00:03:09.891 --> 00:03:14.058
of the absolute value of
e to the third power minus

00:03:16.323 --> 00:03:17.530
15

00:03:17.530 --> 00:03:22.174
times the natural log of
the absolute value of e.

00:03:22.174 --> 00:03:23.967
So what does this simplify to?

00:03:23.967 --> 00:03:25.608
The natural log of e to the third power,

00:03:25.608 --> 00:03:27.148
what power do I have to raise e to,

00:03:27.148 --> 00:03:28.250
to get to e to the third?

00:03:28.250 --> 00:03:29.957
Well, that's just going to be three.

00:03:29.957 --> 00:03:32.815
And then the natural log of e,

00:03:32.815 --> 00:03:35.318
what power do I have to
raise e to, to get e?

00:03:35.318 --> 00:03:36.760
Well, that's just one.

00:03:36.760 --> 00:03:40.288
So this is 15 times three minus 15.

00:03:40.288 --> 00:03:43.371
So that is all going to get us to 30,

00:03:44.761 --> 00:03:47.178
and we are done, 45 minus 15.


Worked example: separable differential equations

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:01.612
- [Instructor] What we're
gonna be doing in this video

00:00:01.612 --> 00:00:04.513
is get some practice
finding general solutions to

00:00:04.513 --> 00:00:07.059
separable differential equations.

00:00:07.059 --> 00:00:10.369
So let's say that I had the
differential equation DY, DX,

00:00:10.369 --> 00:00:13.279
the derivative of Y with
respect to X, is equal to

00:00:13.279 --> 00:00:14.862
E to the X, over Y.

00:00:15.903 --> 00:00:18.747
See if you can find
the general solution to

00:00:18.747 --> 00:00:20.143
this differential equation.

00:00:20.143 --> 00:00:21.245
I'm giving you a huge hint.

00:00:21.245 --> 00:00:24.122
It is a separable differential equation.

00:00:24.122 --> 00:00:25.524
All right, so when we're dealing with

00:00:25.524 --> 00:00:27.530
a separable differential
equation, what we wanna do

00:00:27.530 --> 00:00:30.711
is get the Ys and the
DYs on one side, and then

00:00:30.711 --> 00:00:32.421
the Xs and the DXs on the other side.

00:00:32.421 --> 00:00:34.646
And we really treat these
differentials kind of

00:00:34.646 --> 00:00:36.785
like variables, which
is a little hand-wavy

00:00:36.785 --> 00:00:37.959
with the mathematics.

00:00:37.959 --> 00:00:39.489
But that's what we will do.

00:00:39.489 --> 00:00:44.015
So let's see. If we
multiply both sides times Y,

00:00:44.015 --> 00:00:46.620
so we're gonna multiply
both sides times Y,

00:00:46.620 --> 00:00:47.714
what are we going to get?

00:00:47.714 --> 00:00:51.253
We're gonna get Y times a derivative of Y,

00:00:51.253 --> 00:00:55.497
with respect to X, is equal to E to the X,

00:00:55.497 --> 00:00:58.615
and now we can multiply both
sides by the differential,

00:00:58.615 --> 00:01:03.574
DX; multiply both of them
by DX; those cancel out.

00:01:03.574 --> 00:01:07.741
And we are left with Y times
DY is equal to E to the X, DX.

00:01:12.508 --> 00:01:15.884
And now we can take the
integral of both sides.

00:01:15.884 --> 00:01:17.881
So let us do that.

00:01:17.881 --> 00:01:20.946
So what is the integral of Y, DY?

00:01:20.946 --> 00:01:23.275
Well here we would just
use the reverse power rule.

00:01:23.275 --> 00:01:26.831
We would increment the exponent,
so it's Y to the first,

00:01:26.831 --> 00:01:28.622
but so now when we take
the anti-derivative,

00:01:28.622 --> 00:01:31.649
it will be Y squared, and
then we divide by that

00:01:31.649 --> 00:01:35.499
incremented exponent, is equal to, well

00:01:35.499 --> 00:01:37.630
the exciting thing about
E to the X is that it's

00:01:37.630 --> 00:01:40.983
anti-derivative, and its
derivative, is E to the X,

00:01:40.983 --> 00:01:45.150
is equal to E to the X,
plus is equal to E to the X

00:01:47.262 --> 00:01:48.720
plus C.

00:01:48.720 --> 00:01:52.190
And so we can leave it
like this if we like.

00:01:52.190 --> 00:01:54.120
In fact this right over
here is, this isn't

00:01:54.120 --> 00:01:55.626
an explicit function.

00:01:55.626 --> 00:01:58.672
Y here isn't an explicit function of X.

00:01:58.672 --> 00:02:01.633
We could actually say Y is
equal to the plus or minus

00:02:01.633 --> 00:02:05.168
square root of two times
all of this business,

00:02:05.168 --> 00:02:09.455
but this would be a pretty
general relationship,

00:02:09.455 --> 00:02:13.657
which would satisfy this
separable differential equation.

00:02:13.657 --> 00:02:16.347
Let's do another example.

00:02:16.347 --> 00:02:20.097
So let's say that we
have the derivative of Y

00:02:21.549 --> 00:02:26.314
with respect to X is equal
to, let's say it's equal to

00:02:26.314 --> 00:02:28.481
Y squared times sine of X.

00:02:30.609 --> 00:02:32.491
Pause the video and see if you can find

00:02:32.491 --> 00:02:34.421
the general solution here.

00:02:34.421 --> 00:02:37.549
So once again, we wanna
separate our Ys and our Xs.

00:02:37.549 --> 00:02:40.225
So let's see, we can
multiply both sides times

00:02:40.225 --> 00:02:43.836
Y to the negative two power,
Y to the negative two,

00:02:43.836 --> 00:02:47.422
Y to the negative two, these become one,

00:02:47.422 --> 00:02:50.501
and then we could also
multiply both sides times DX.

00:02:50.501 --> 00:02:53.623
So if we multiply DX here,
those cancel out, and then

00:02:53.623 --> 00:02:56.647
we multiply DX here,
and so we're left with

00:02:56.647 --> 00:03:01.102
Y to the negative two
power times DY is equal to

00:03:01.102 --> 00:03:05.269
sine of X, DX, and now we
just can integrate both sides.

00:03:08.524 --> 00:03:11.790
Now what is the anti-derivative
of Y to the negative two?

00:03:11.790 --> 00:03:15.413
Well, once again we use
the reverse power rule.

00:03:15.413 --> 00:03:17.485
We increment the exponents,
so it's gonna be Y

00:03:17.485 --> 00:03:19.525
to the negative one, and
then we divide by that

00:03:19.525 --> 00:03:21.169
newly incremented exponent.

00:03:21.169 --> 00:03:23.011
So we divide by negative one.

00:03:23.011 --> 00:03:25.001
Well that would just
make this think negative.

00:03:25.001 --> 00:03:27.584
That is going to be equal to...

00:03:28.761 --> 00:03:32.440
So, what's the
anti-derivative of sine of X?

00:03:32.440 --> 00:03:34.398
Well, it is, you might recognize it if

00:03:34.398 --> 00:03:36.629
I put a negative there,
and a negative there.

00:03:36.629 --> 00:03:38.293
The anti-derivative of negative sine of X,

00:03:38.293 --> 00:03:39.557
well that's cosine of X.

00:03:39.557 --> 00:03:43.838
So this whole thing is gonna
be negative cosine of X,

00:03:43.838 --> 00:03:46.469
or another way to write this:
I could multiply both sides

00:03:46.469 --> 00:03:48.519
times a negative one, and so these would

00:03:48.519 --> 00:03:51.609
both become positive, and so I could write

00:03:51.609 --> 00:03:55.609
one over Y is equal to
cosine of X, and actually

00:03:57.728 --> 00:03:59.444
let me write it this
way, plus C; don't wanna

00:03:59.444 --> 00:04:00.947
forget my plus Cs.

00:04:00.947 --> 00:04:04.492
Plus C, or I could take the
reciprocal of both sides

00:04:04.492 --> 00:04:07.394
if I wanna solve explicitly
for Y, I could get Y

00:04:07.394 --> 00:04:10.644
is equal to one over cosine of X plus C

00:04:15.388 --> 00:04:17.388
as our general solution.

00:04:18.283 --> 00:04:19.993
And we're done.

00:04:19.993 --> 00:04:21.993
That was strangely fun.


Integration by parts: definite integrals

WEBVTT
Kind: captions
Language: en

00:00:00.672 --> 00:00:01.681
- [Instructor] We're
gonna do in this video

00:00:01.681 --> 00:00:04.243
is try to evaluate the
definite integral from zero

00:00:04.243 --> 00:00:06.910
to pi of x cosine of x dx.

00:00:10.276 --> 00:00:12.421
Like always, pause this video and see

00:00:12.421 --> 00:00:15.236
if you can evaluate it yourself.

00:00:15.236 --> 00:00:16.437
Well when you immediately look at this,

00:00:16.437 --> 00:00:18.459
it's not obvious how you
just straight up take

00:00:18.459 --> 00:00:21.230
the anti-derivative here
and then evaluate that

00:00:21.230 --> 00:00:23.314
at pi and then subtract from that

00:00:23.314 --> 00:00:25.933
and evaluate it at zero, so
we're probably going to have

00:00:25.933 --> 00:00:29.003
to use a slightly more
sophisticated technique.

00:00:29.003 --> 00:00:33.215
And in general, if you
see a product of functions

00:00:33.215 --> 00:00:35.721
right over here, and if
one of these functions

00:00:35.721 --> 00:00:39.114
is fairly straightforward to
take the anti-derivative of

00:00:39.114 --> 00:00:42.808
without making it more
complicated like cosine of x,

00:00:42.808 --> 00:00:46.184
and another of the functions
like x if you were to take

00:00:46.184 --> 00:00:48.116
its derivative, it gets simpler.

00:00:48.116 --> 00:00:49.737
In this case, it would just become one.

00:00:49.737 --> 00:00:51.847
It's a pretty good sign
that we should be using

00:00:51.847 --> 00:00:54.353
integration by parts.

00:00:54.353 --> 00:00:58.551
So let's just remind ourselves
about integration by parts.

00:00:58.551 --> 00:01:01.617
So integration by parts,
I'll do it right over here,

00:01:01.617 --> 00:01:04.446
if I have the integral
and I'll just write this

00:01:04.446 --> 00:01:06.597
as an indefinite integral
but here we wanna take

00:01:06.597 --> 00:01:09.066
the indefinite integral
and then evaluate it at pi

00:01:09.066 --> 00:01:12.829
and evaluate it at zero,
so if I have f of x

00:01:12.829 --> 00:01:14.746
times g prime of x, dx,

00:01:19.908 --> 00:01:22.300
this is going to be equal
to, and in other videos

00:01:22.300 --> 00:01:23.879
we prove this, it really
just comes straight

00:01:23.879 --> 00:01:25.771
out of the product rule that you learned

00:01:25.771 --> 00:01:27.761
in differential calculus,
this is gonna be equal

00:01:27.761 --> 00:01:31.928
to f of x times g of x minus,
you then swap these around,

00:01:34.638 --> 00:01:37.221
minus f prime of x, g of x, dx.

00:01:40.708 --> 00:01:43.061
And just to reiterate what I said before,

00:01:43.061 --> 00:01:47.089
you wanna find an f of x that
when I take its derivative,

00:01:47.089 --> 00:01:49.589
it simplifies it, so simplify,

00:01:52.082 --> 00:01:56.336
and you wanna find a g
prime of x that when I take

00:01:56.336 --> 00:02:00.336
its anti-derivative, so when
I take its anti-derivative,

00:02:00.336 --> 00:02:02.618
it doesn't get more complicated.

00:02:02.618 --> 00:02:04.618
So not more complicated.

00:02:07.387 --> 00:02:09.757
Because if the f of x gets
simplified when I take

00:02:09.757 --> 00:02:12.303
its derivative, and the
g prime of x does not

00:02:12.303 --> 00:02:15.217
get more complicated when
I take its anti-derivative,

00:02:15.217 --> 00:02:18.348
then this expression will
maybe be easier to find

00:02:18.348 --> 00:02:20.669
the anti-derivative of.

00:02:20.669 --> 00:02:22.261
So let's do that over here.

00:02:22.261 --> 00:02:24.248
Between x and cosine of x, which one

00:02:24.248 --> 00:02:28.076
gets more simple when
I take its derivative?

00:02:28.076 --> 00:02:31.322
Well the derivative of x is
just one, so I'm gonna make that

00:02:31.322 --> 00:02:35.631
my f of x, so I could
write that over here,

00:02:35.631 --> 00:02:39.686
so my f of x, I will
say is x, in which case,

00:02:39.686 --> 00:02:43.853
f prime of x, f prime of x,
is going to be equal to one,

00:02:46.395 --> 00:02:49.773
and then what would my g prime of x be?

00:02:49.773 --> 00:02:52.828
Well my g prime of x,
cosine of x if I take

00:02:52.828 --> 00:02:55.322
its anti-derivative, it
doesn't get more complicated.

00:02:55.322 --> 00:02:57.857
The anti-derivative of
cosine of x is sine of x.

00:02:57.857 --> 00:03:00.173
So let me make that my g prime of x.

00:03:00.173 --> 00:03:03.896
So g prime of x is equal to cosine of x

00:03:03.896 --> 00:03:07.515
in which case g of x, the
anti-derivative of cosine of x,

00:03:07.515 --> 00:03:10.284
well it's just sine of x or
another way to think about it,

00:03:10.284 --> 00:03:12.426
the derivative of sine
of x is cosine of x.

00:03:12.426 --> 00:03:14.801
Now you could think about
plus C's and all of that

00:03:14.801 --> 00:03:16.295
but remember, this is gonna
be a definite integral

00:03:16.295 --> 00:03:18.589
so all of those arbitrary
constants are going to get

00:03:18.589 --> 00:03:20.387
canceled out.

00:03:20.387 --> 00:03:21.842
So now let's think through this.

00:03:21.842 --> 00:03:26.487
Let's just apply the
integration by parts here.

00:03:26.487 --> 00:03:29.343
In this particular case,
all of this is going to be

00:03:29.343 --> 00:03:31.821
equal to, so I'm saying
that is equal to this,

00:03:31.821 --> 00:03:34.269
I'm gonna skip down here,
it's going to be equal

00:03:34.269 --> 00:03:36.558
to f of x times g of x.

00:03:36.558 --> 00:03:41.458
So that is f of x is
x, g of x is sine of x,

00:03:41.458 --> 00:03:44.958
f of x times g of x, minus the integral

00:03:47.334 --> 00:03:51.278
of f prime of x, f prime of x is just one,

00:03:51.278 --> 00:03:54.488
we could write it like
that, one times g of x,

00:03:54.488 --> 00:03:58.553
g of x is sine of x, so we
could write it like this,

00:03:58.553 --> 00:04:00.986
but one times sine of x, well
we could just rewrite that

00:04:00.986 --> 00:04:03.831
as sine of x, it'll make
it a little bit simpler,

00:04:03.831 --> 00:04:07.998
sine of x, dx, and then
remember, this is a definite

00:04:09.207 --> 00:04:11.815
integral, so we are
going to want to evaluate

00:04:11.815 --> 00:04:15.529
this whole thing at pi
and at zero, and then take

00:04:15.529 --> 00:04:17.795
the difference between the two.

00:04:17.795 --> 00:04:21.756
But what is the indefinite
integral of sine of x dx?

00:04:21.756 --> 00:04:25.316
Well, or to say the anti-derivative of it,

00:04:25.316 --> 00:04:29.181
we know that the derivative
of cosine is negative sine

00:04:29.181 --> 00:04:31.505
of x, and so in fact what
we want, we could bring

00:04:31.505 --> 00:04:33.912
this negative sine into the
integral, so we could say

00:04:33.912 --> 00:04:36.830
plus the integral of negative sine of x,

00:04:36.830 --> 00:04:39.227
now this clearly the anti-derivative here

00:04:39.227 --> 00:04:42.585
is cosine of x, so this
thing is going to be

00:04:42.585 --> 00:04:45.531
cosine of x, and now we
just have to evaluate it

00:04:45.531 --> 00:04:47.683
at the end points.

00:04:47.683 --> 00:04:50.936
So let's first evaluate
this whole thing at pi.

00:04:50.936 --> 00:04:54.916
So this is going to be
equal to pi sine of pi,

00:04:54.916 --> 00:04:57.666
pi sine of pi, plus cosine of pi,

00:05:00.815 --> 00:05:04.354
and then from that I'm going
to subtract this whole thing

00:05:04.354 --> 00:05:08.521
evaluated at zero, so let me
do zero in a different color,

00:05:09.459 --> 00:05:13.135
at zero, so it's going
to be zero times sine

00:05:13.135 --> 00:05:16.635
of zero plus cosine of zero, so let's see,

00:05:17.689 --> 00:05:20.540
sine of pi is just zero
so this is just going to

00:05:20.540 --> 00:05:21.538
cancel out.

00:05:21.538 --> 00:05:25.101
Cosine of pi, that is negative one,

00:05:25.101 --> 00:05:28.319
and then this is zero,
and then cosine of zero,

00:05:28.319 --> 00:05:32.668
that is one, so you have
negative one minus one,

00:05:32.668 --> 00:05:35.213
so this all gets us to negative two,

00:05:35.213 --> 00:05:36.957
and we are done.

00:05:36.957 --> 00:05:39.372
Using integration by parts,
we were able to evaluate

00:05:39.372 --> 00:05:41.289
this definite integral.


WEBVTT
Kind: captions
Language: en

00:00:00.869 --> 00:00:05.017
- [Salman] Welcome to the Khan
Academy 2016 Annual Report.

00:00:05.017 --> 00:00:06.385
In the actual text of the report,

00:00:06.385 --> 00:00:08.138
we're going to go into a lot more detail

00:00:08.138 --> 00:00:10.138
on the financials and other things,

00:00:10.138 --> 00:00:12.534
but I'm hoping here to
give you an overview.

00:00:12.534 --> 00:00:13.677
Big picture,

00:00:13.677 --> 00:00:16.927
2016 was a great year for Khan Academy.

00:00:17.803 --> 00:00:20.339
The impact has continued to grow

00:00:20.339 --> 00:00:23.534
in the double-digit percentages
and is accelerating.

00:00:23.534 --> 00:00:26.170
So one of the areas that
we've really started

00:00:26.170 --> 00:00:30.430
to accelerate in 2016 was
in content development.

00:00:30.430 --> 00:00:33.230
Many people associate Khan
Academy with mathematics,

00:00:33.230 --> 00:00:34.901
but what we've really been investing in

00:00:34.901 --> 00:00:36.643
is going beyond mathematics.

00:00:36.643 --> 00:00:39.488
We really want to cover all
of the core academic subjects

00:00:39.488 --> 00:00:41.557
that you consider to be
part of your education,

00:00:41.557 --> 00:00:44.213
all the way through the core of college.

00:00:44.213 --> 00:00:46.741
And it's not just videos,
we're doing a lot on practice,

00:00:46.741 --> 00:00:49.447
and we're doing a lot of written articles.

00:00:49.447 --> 00:00:51.957
2016 was a very big deal

00:00:51.957 --> 00:00:54.945
in terms of a partnership
with the College Board.

00:00:54.945 --> 00:00:56.826
A few years ago, they reached out to us,

00:00:56.826 --> 00:00:59.190
said, "We are going to revamp the SAT,

00:00:59.190 --> 00:01:01.855
"and as part of that we
want to address at least

00:01:01.855 --> 00:01:04.873
"the perception of inequity
that's been around for decades."

00:01:04.873 --> 00:01:07.411
And the reason why they wanted
to partner with Khan Academy

00:01:07.411 --> 00:01:10.950
is traditional test prep was
associated with gaming an exam.

00:01:10.950 --> 00:01:12.680
It wasn't about real learning.

00:01:12.680 --> 00:01:15.289
But they saw that we were
focused on filling in gaps

00:01:15.289 --> 00:01:16.725
in students understanding,

00:01:16.725 --> 00:01:18.753
so we launched with the College Board.

00:01:18.753 --> 00:01:21.362
And what we've seen over
the course of the year is

00:01:21.362 --> 00:01:23.272
Khan Academy is now used more than

00:01:23.272 --> 00:01:25.775
all of paid test prep combined,

00:01:25.775 --> 00:01:29.594
and we're seeing that usage
very broad and very deep.

00:01:29.594 --> 00:01:32.480
And what's been even more
exciting is the PSAT,

00:01:32.480 --> 00:01:34.645
which is taken in 10th or 11th grade

00:01:34.645 --> 00:01:37.961
by 80% of American high school students.

00:01:37.961 --> 00:01:40.989
In the past, that was just a
random practice test you took,

00:01:40.989 --> 00:01:43.838
but now it acts as a
diagnostic for Khan Academy.

00:01:43.838 --> 00:01:45.652
You can take your PSAT scores,

00:01:45.652 --> 00:01:47.514
give permission to Khan Academy,

00:01:47.514 --> 00:01:48.924
and then Khan Academy software

00:01:48.924 --> 00:01:50.257
knows your strengths and weaknesses,

00:01:50.257 --> 00:01:53.263
not just in math but in
reading and writing as well,

00:01:53.263 --> 00:01:55.469
and can immediately
give you practice items

00:01:55.469 --> 00:01:57.640
that are essentially weak point training.

00:01:57.640 --> 00:02:00.658
And even more powerful, we've
been able to follow students

00:02:00.658 --> 00:02:04.078
from that PSAT through
their work on Khan Academy

00:02:04.078 --> 00:02:06.087
to eventually taking the SAT,

00:02:06.087 --> 00:02:10.160
to understand the efficacy
of this entire ecosystem.

00:02:10.160 --> 00:02:12.148
And what you see here on this slide

00:02:12.148 --> 00:02:14.638
are some of the preliminary
results for arguably

00:02:14.638 --> 00:02:17.750
the largest personalized
learning study ever conducted.

00:02:17.750 --> 00:02:19.796
With over 300,000 students,

00:02:19.796 --> 00:02:22.748
we saw that those who
prepared on Khan Academy

00:02:22.748 --> 00:02:26.169
for at least 20 hours saw twice the growth

00:02:26.169 --> 00:02:29.531
from the PSAT to the SAT than
they would typically see,

00:02:29.531 --> 00:02:31.159
and this is just the beginning.

00:02:31.159 --> 00:02:33.161
We're working with the
College Board very closely

00:02:33.161 --> 00:02:34.917
to do even further studies,

00:02:34.917 --> 00:02:37.974
and we've also seen that it
doesn't stop at 20 hours.

00:02:37.974 --> 00:02:40.358
The more time that
students put in practice,

00:02:40.358 --> 00:02:42.477
the more gains were seen.

00:02:42.477 --> 00:02:45.548
Another really exciting event in 2016

00:02:45.548 --> 00:02:48.567
is that Duck Duck Moose
became part of Khan Academy.

00:02:48.567 --> 00:02:50.632
Duck Duck Moose, I would argue,

00:02:50.632 --> 00:02:53.564
was the best early learning
app company out there.

00:02:53.564 --> 00:02:56.035
In 2016, the founding team

00:02:56.035 --> 00:02:58.986
said, "We would rather see our
work having a social mission

00:02:58.986 --> 00:03:01.001
"and reaching as many
students as possible,

00:03:01.001 --> 00:03:04.020
"especially those that are most in need."

00:03:04.020 --> 00:03:06.890
And so they became part
of our team in 2016.

00:03:06.890 --> 00:03:07.967
And going forward,

00:03:07.967 --> 00:03:10.614
we're going to be working
on early learning apps

00:03:10.614 --> 00:03:14.365
that holistically approach
reading, writing, mathematics,

00:03:14.365 --> 00:03:16.063
starting at pre-kindergarten

00:03:16.063 --> 00:03:20.195
and then taking students
through early elementary school.

00:03:20.195 --> 00:03:21.926
International continues to be

00:03:21.926 --> 00:03:23.881
a very big part of Khan Academy.

00:03:23.881 --> 00:03:28.113
Roughly 1/3 of our usage is
outside of the United States.

00:03:28.113 --> 00:03:32.729
We have increased our focus
on India, Mexico, and Brazil.

00:03:32.729 --> 00:03:35.906
And going forward, we're
building tools and platforms

00:03:35.906 --> 00:03:38.944
to allow each geography
to be even more localized

00:03:38.944 --> 00:03:41.246
to their specific needs.

00:03:41.246 --> 00:03:44.477
And going forward, there's
a lot of exciting things.

00:03:44.477 --> 00:03:45.929
What you see here is a screenshot

00:03:45.929 --> 00:03:47.778
for the student-facing experience

00:03:47.778 --> 00:03:50.051
for back to school 2017,

00:03:50.051 --> 00:03:51.980
where students can tell Khan Academy

00:03:51.980 --> 00:03:53.610
what classes they're taking,

00:03:53.610 --> 00:03:55.061
what grade they're in,

00:03:55.061 --> 00:03:57.518
and then Khan Academy
can personalize to them.

00:03:57.518 --> 00:03:59.407
And also, I have game mechanics

00:03:59.407 --> 00:04:02.498
so that the students are
motivated to move forward.

00:04:02.498 --> 00:04:04.420
You can see that right over here,

00:04:04.420 --> 00:04:07.101
once you get into a unit page.

00:04:07.101 --> 00:04:08.809
Of course, you see the videos.

00:04:08.809 --> 00:04:10.790
You see the practice items here.

00:04:10.790 --> 00:04:13.663
But we are also adding diagnostic quizzes

00:04:13.663 --> 00:04:16.180
and unit tests as well.

00:04:16.180 --> 00:04:18.800
What you see here is our
assignments functionality,

00:04:18.800 --> 00:04:20.434
where a teacher can assign any piece

00:04:20.434 --> 00:04:22.169
of content on Khan Academy.

00:04:22.169 --> 00:04:24.950
This is a teacher assigning
some practice content.

00:04:24.950 --> 00:04:26.460
They can assign it to particular students

00:04:26.460 --> 00:04:27.750
in a particular class.

00:04:27.750 --> 00:04:30.187
They can set a due date.

00:04:30.187 --> 00:04:31.539
They can then see a readout

00:04:31.539 --> 00:04:36.169
of which students completed
which assignments.

00:04:36.169 --> 00:04:37.734
And then they can see

00:04:37.734 --> 00:04:40.890
which items gave the
students the most difficulty

00:04:40.890 --> 00:04:43.903
so that they can review
it in class the next day.

00:04:43.903 --> 00:04:45.348
On top of all of this,

00:04:45.348 --> 00:04:47.297
we have a program called LearnStorm,

00:04:47.297 --> 00:04:48.729
which is a learning challenge

00:04:48.729 --> 00:04:51.396
that schools, teachers,
students participate in

00:04:51.396 --> 00:04:53.588
that celebrate the growth mindset.

00:04:53.588 --> 00:04:56.066
The more students can
show grit, perseverance,

00:04:56.066 --> 00:04:58.521
as they work through
practice on Khan Academy,

00:04:58.521 --> 00:05:00.509
the more awards and recognition

00:05:00.509 --> 00:05:02.682
the students, their
classrooms, their teachers,

00:05:02.682 --> 00:05:04.246
and their schools will get.

00:05:04.246 --> 00:05:06.977
We've piloted it in several
geographies in the past,

00:05:06.977 --> 00:05:08.919
and 2017 is going to be the first time

00:05:08.919 --> 00:05:12.252
that we do this on a nationwide scale.

00:05:12.252 --> 00:05:14.431
So there's a lot going on at Khan Academy.

00:05:14.431 --> 00:05:17.667
What I just covered is
really just a sample of it.

00:05:17.667 --> 00:05:20.831
But we're really grateful
for your support.

00:05:20.831 --> 00:05:22.178
We have a big mission,

00:05:22.178 --> 00:05:25.822
a free, world-class education
for anyone, anywhere.

00:05:25.822 --> 00:05:29.594
And some of y'all have been
with us for many years.

00:05:29.594 --> 00:05:34.230
When we were just starting in
2007, out of a walk-in closet,

00:05:34.230 --> 00:05:37.524
this type of a mission would've
seemed, frankly, delusional.

00:05:37.524 --> 00:05:40.337
But because of your
support, your partnership,

00:05:40.337 --> 00:05:42.476
all of the work that we've
been able to do together

00:05:42.476 --> 00:05:44.914
with governments, with
teachers, with students,

00:05:44.914 --> 00:05:47.111
it's no longer just a dream.

00:05:47.111 --> 00:05:48.525
It's really happening.

00:05:48.525 --> 00:05:51.396
And the momentum that we've
built together in 2016

00:05:51.396 --> 00:05:54.454
is continuing to accelerate into 2017,

00:05:54.454 --> 00:05:57.961
and I expect is going to
accelerate in 2018 and beyond

00:05:57.961 --> 00:06:00.899
so that, as we look
back in a decade or two,

00:06:00.899 --> 00:06:03.508
we'll say how could we
ever imagine a world

00:06:03.508 --> 00:06:06.717
where there wasn't a free,
world-class education

00:06:06.717 --> 00:06:08.467
for anyone, anywhere?


Rewriting before integrating

WEBVTT
Kind: captions
Language: en

00:00:00.649 --> 00:00:01.887
- [Instructor] Let's say
that we wanted to take

00:00:01.887 --> 00:00:05.007
the indefinite integral of X squared

00:00:05.007 --> 00:00:07.257
times three X minus one DX.

00:00:08.778 --> 00:00:11.720
Pause this video and see
if you can evaluate this.

00:00:11.720 --> 00:00:15.231
So, you might be saying oh,
what kind of fancy technique

00:00:15.231 --> 00:00:16.383
could I use?

00:00:16.383 --> 00:00:18.679
But you will see sometimes the fanciest

00:00:18.679 --> 00:00:21.079
or maybe the least fancy
but the best technique

00:00:21.079 --> 00:00:23.915
is to just simplify this algebraically.

00:00:23.915 --> 00:00:25.775
So, in this situation what would happens

00:00:25.775 --> 00:00:29.251
if we distribute this X squared?

00:00:29.251 --> 00:00:32.285
Well, then we're going
to get a polynomial here

00:00:32.285 --> 00:00:33.640
within the integral,

00:00:33.640 --> 00:00:35.719
so this is going to be
equal to the integral

00:00:35.719 --> 00:00:37.886
of X squared times three X

00:00:38.889 --> 00:00:40.806
is three X to the third

00:00:43.159 --> 00:00:48.127
and then negative one times
X squared is minus X squared

00:00:48.127 --> 00:00:49.960
and then that times DX

00:00:51.548 --> 00:00:54.541
and now this is pretty
straightforward to evaluate.

00:00:54.541 --> 00:00:58.538
This is going to be equal
to the anti-derivative

00:00:58.538 --> 00:01:02.363
of X to the third is X
to the fourth over four

00:01:02.363 --> 00:01:06.781
so this is going to be
three times X to the fourth

00:01:06.781 --> 00:01:09.057
over four, I could write it that way

00:01:09.057 --> 00:01:13.134
or let me just write it
X to the fourth over four

00:01:13.134 --> 00:01:16.023
and then the anti-derivative of X squared

00:01:16.023 --> 00:01:18.136
is X to the third over three,

00:01:18.136 --> 00:01:21.223
so minus X to the third over three

00:01:21.223 --> 00:01:22.472
and this is an indefinite integral,

00:01:22.472 --> 00:01:24.454
there might be a constant there,

00:01:24.454 --> 00:01:27.225
so let me write that down and we're done.

00:01:27.225 --> 00:01:29.024
The big takeaway is you just have to do

00:01:29.024 --> 00:01:32.347
a little bit of distribution to get a form

00:01:32.347 --> 00:01:36.043
where it's easier to
evaluate the anti-derivative.

00:01:36.043 --> 00:01:37.847
Let's do another example.

00:01:37.847 --> 00:01:39.481
Let's say that we want to take

00:01:39.481 --> 00:01:41.954
the indefinite integral of,

00:01:41.954 --> 00:01:43.865
this is gonna be a hairy expression,

00:01:43.865 --> 00:01:48.032
so X to the third plus
three X squared minus five,

00:01:49.001 --> 00:01:51.501
all of that over X squared DX.

00:01:54.609 --> 00:01:55.551
What would this be?

00:01:55.551 --> 00:01:56.432
Pause the video again

00:01:56.432 --> 00:01:59.232
and see if you can figure it out.

00:01:59.232 --> 00:02:01.302
So, once again your brain
might want to try to do

00:02:01.302 --> 00:02:03.577
some fancy tricks or whatever else

00:02:03.577 --> 00:02:06.080
but the main insight here

00:02:06.080 --> 00:02:08.608
is to realize that you could
just simplify it algebraically.

00:02:08.608 --> 00:02:11.158
What happens if you just
divide each of these terms

00:02:11.158 --> 00:02:12.348
by X squared?

00:02:12.348 --> 00:02:15.287
Well, then this thing
is going to be equal to,

00:02:15.287 --> 00:02:16.543
that's in parentheses here,

00:02:16.543 --> 00:02:19.364
X to the third divided by X squared

00:02:19.364 --> 00:02:21.358
is just going to be X.

00:02:21.358 --> 00:02:24.109
Three X squared divided by X squared

00:02:24.109 --> 00:02:25.593
is just three

00:02:25.593 --> 00:02:28.768
and then negative five
divided by X squared,

00:02:28.768 --> 00:02:30.761
you could just write that as negative five

00:02:30.761 --> 00:02:33.842
times X to the negative two power.

00:02:33.842 --> 00:02:36.719
And so, once again we just need to use

00:02:36.719 --> 00:02:38.792
the reverse power rule here

00:02:38.792 --> 00:02:40.894
to take the anti-derivative.

00:02:40.894 --> 00:02:41.779
This is going to be,

00:02:41.779 --> 00:02:43.206
let's see the anti-derivative of X

00:02:43.206 --> 00:02:45.049
is X squared over two,

00:02:45.049 --> 00:02:50.037
X squared over two plus the
anti-derivative of three

00:02:50.037 --> 00:02:51.909
would just be three X.

00:02:51.909 --> 00:02:54.646
The anti-derivative of negative five X

00:02:54.646 --> 00:02:56.157
to the negative two.

00:02:56.157 --> 00:02:58.907
So, we would increment
the exponent by one,

00:02:58.907 --> 00:03:01.918
positive one and then
divide by that value.

00:03:01.918 --> 00:03:06.085
So, it would be negative
five X to the negative one,

00:03:08.549 --> 00:03:10.364
we're adding one to the negative one,

00:03:10.364 --> 00:03:13.251
all of that divided by negative one

00:03:13.251 --> 00:03:14.084
which is the same,

00:03:14.084 --> 00:03:15.463
we could write it like that.

00:03:15.463 --> 00:03:17.060
Well, these two would just,

00:03:17.060 --> 00:03:18.212
you would have minus

00:03:18.212 --> 00:03:19.937
and then you're dividing by negative one,

00:03:19.937 --> 00:03:21.192
so it's really just going,

00:03:21.192 --> 00:03:23.925
we can rewrite it like this
plus five X to the negative one

00:03:23.925 --> 00:03:25.084
and you could take the derivative of this

00:03:25.084 --> 00:03:27.794
to verify that it would
indeed give you that

00:03:27.794 --> 00:03:30.485
and of course we can't forget our plus C,

00:03:30.485 --> 00:03:31.482
never forget that

00:03:31.482 --> 00:03:34.425
if you're taking an indefinite integral.

00:03:34.425 --> 00:03:38.077
Alright, let's just do
one more for good measure.

00:03:38.077 --> 00:03:40.468
Let's say we're taking
the indefinite integral

00:03:40.468 --> 00:03:43.635
of the cube root of X to the fifth DX.

00:03:48.437 --> 00:03:51.183
Pause the video and see
if you can evaluate this.

00:03:51.183 --> 00:03:52.883
Try to write it a little bit neater.

00:03:52.883 --> 00:03:54.909
X to the fifth DX.

00:03:54.909 --> 00:03:57.449
Pause the video and try to figure it out.

00:03:57.449 --> 00:03:59.080
So, here the realization is well,

00:03:59.080 --> 00:04:01.744
if you just rewrite all
this as one exponent,

00:04:01.744 --> 00:04:04.858
so this is equal to
the indefinite integral

00:04:04.858 --> 00:04:07.775
of X to the fifth to the one third,

00:04:09.009 --> 00:04:11.004
I just rewrote the cube root

00:04:11.004 --> 00:04:13.609
as the one third power DX,

00:04:13.609 --> 00:04:15.429
which is the same thing as the integral

00:04:15.429 --> 00:04:18.143
of X to the, if I raise
something to a power

00:04:18.143 --> 00:04:19.151
and then raise that to a power

00:04:19.151 --> 00:04:20.906
I can multiply those two exponents,

00:04:20.906 --> 00:04:22.572
that's just exponent properties,

00:04:22.572 --> 00:04:24.542
X to the five thirds DX,

00:04:24.542 --> 00:04:27.283
many of you might have just
gone straight to this step

00:04:27.283 --> 00:04:28.325
right over here

00:04:28.325 --> 00:04:29.325
and then once again,

00:04:29.325 --> 00:04:31.997
we just have to use
the reverse power rule.

00:04:31.997 --> 00:04:34.954
This is going to be X to the,

00:04:34.954 --> 00:04:37.539
we increment this five thirds by one

00:04:37.539 --> 00:04:39.637
or we can add three thirds to it,

00:04:39.637 --> 00:04:41.965
so it's X to the eight thirds

00:04:41.965 --> 00:04:43.557
and then we divide by eight thirds

00:04:43.557 --> 00:04:45.122
or multiply by its reciprocal

00:04:45.122 --> 00:04:47.943
so we could just say three eighths

00:04:47.943 --> 00:04:49.751
times X to the eight thirds

00:04:49.751 --> 00:04:52.143
and of course, we have our
plus C and verify this.

00:04:52.143 --> 00:04:53.405
If you use the power rule here

00:04:53.405 --> 00:04:55.048
you'd have eight thirds
times the three eights,

00:04:55.048 --> 00:04:57.060
will just give you a coefficient of one

00:04:57.060 --> 00:04:59.881
and then you decrement this
by three thirds or one,

00:04:59.881 --> 00:05:01.105
you get to five thirds

00:05:01.105 --> 00:05:04.012
which is exactly what we originally had.

00:05:04.012 --> 00:05:06.647
So, the big takeaway of this video,

00:05:06.647 --> 00:05:10.002
many times the most powerful
integration technique

00:05:10.002 --> 00:05:14.169
is literally just algebraic
simplification first.


Worked example: Rewriting limit of Riemann sum as definite integral

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.395
- [Instructor] So,
we've got a Riemann sum.

00:00:02.395 --> 00:00:04.925
We're gonna take the limit
as N approaches infinity

00:00:04.925 --> 00:00:06.157
and the goal of this video

00:00:06.157 --> 00:00:08.137
is to see if we can rewrite this

00:00:08.137 --> 00:00:09.756
as a definite integral.

00:00:09.756 --> 00:00:11.176
I encourage you to pause the video

00:00:11.176 --> 00:00:14.775
and see if you can work
through it on your own.

00:00:14.775 --> 00:00:16.054
So, let's remind ourselves

00:00:16.054 --> 00:00:20.428
how a definite integral can
relate to a Riemann sum.

00:00:20.428 --> 00:00:24.345
So, if I have the definite
integral from A to B

00:00:27.287 --> 00:00:29.120
of F of X, F of X, DX,

00:00:34.052 --> 00:00:36.391
we have seen in other videos

00:00:36.391 --> 00:00:38.900
this is going to be the limit

00:00:38.900 --> 00:00:43.067
as N approaches infinity
of the sum, capital sigma,

00:00:44.743 --> 00:00:47.076
going from I equals one to N

00:00:47.918 --> 00:00:49.566
and so, essentially
we're gonna sum the areas

00:00:49.566 --> 00:00:51.720
of a bunch of rectangles

00:00:51.720 --> 00:00:55.093
where the width of each
of those rectangles

00:00:55.093 --> 00:00:57.260
we can write as a delta X,

00:00:58.142 --> 00:01:00.916
so your width is going to be delta X

00:01:00.916 --> 00:01:02.777
of each of those rectangles

00:01:02.777 --> 00:01:03.736
and then your height

00:01:03.736 --> 00:01:06.292
is going to be the value of the function

00:01:06.292 --> 00:01:08.434
evaluated some place in that delta X.

00:01:08.434 --> 00:01:10.128
If we're doing a right Riemann sum

00:01:10.128 --> 00:01:12.799
we would do the right
end of that rectangle

00:01:12.799 --> 00:01:14.412
or of that sub interval

00:01:14.412 --> 00:01:18.580
and so, we would start
at our lower bound A

00:01:18.580 --> 00:01:22.747
and we would add as many delta
Xs as our index specifies.

00:01:23.775 --> 00:01:25.198
So, if I is equal to one,

00:01:25.198 --> 00:01:26.942
we add one delta X,

00:01:26.942 --> 00:01:28.962
so we would be at the right
of the first rectangle.

00:01:28.962 --> 00:01:31.356
If I is equal two, we add two delta Xs.

00:01:31.356 --> 00:01:34.630
So, this is going to be delta X

00:01:34.630 --> 00:01:35.963
times our index.

00:01:37.058 --> 00:01:38.623
So, this is the general form

00:01:38.623 --> 00:01:40.649
that we have seen before

00:01:40.649 --> 00:01:42.383
and so, one possibility, you
could even do a little bit

00:01:42.383 --> 00:01:44.373
of pattern matching right here,

00:01:44.373 --> 00:01:47.406
our function looks like
the natural log function,

00:01:47.406 --> 00:01:49.129
so that looks like our func F of X,

00:01:49.129 --> 00:01:51.952
it's the natural log function,

00:01:51.952 --> 00:01:53.330
so I could write that,

00:01:53.330 --> 00:01:56.830
so F of X looks like the natural log of X.

00:01:58.463 --> 00:02:00.079
What else do we see?

00:02:00.079 --> 00:02:02.496
Well, A, that looks like two.

00:02:03.583 --> 00:02:05.575
A is equal to two.

00:02:05.575 --> 00:02:08.110
What would our delta X be?

00:02:08.110 --> 00:02:10.572
Well, you can see this right over here,

00:02:10.572 --> 00:02:12.368
this thing that we're multiplying

00:02:12.368 --> 00:02:14.572
that just is divided by N

00:02:14.572 --> 00:02:17.191
and it's not multiplying by an I,

00:02:17.191 --> 00:02:19.582
this looks like our delta X

00:02:19.582 --> 00:02:22.798
and this right over here
looks like delta X times I.

00:02:22.798 --> 00:02:26.965
So, it looks like our delta
X is equal to five over N.

00:02:28.275 --> 00:02:30.816
So, what can we tell so far?

00:02:30.816 --> 00:02:33.469
Well, we could say that,
okay, this thing up here,

00:02:33.469 --> 00:02:36.660
up the original thing
is going to be equal to

00:02:36.660 --> 00:02:38.109
the definite integral,

00:02:38.109 --> 00:02:41.089
we know our lower bound is going from two

00:02:41.089 --> 00:02:43.148
to we haven't figured
out our upper bound yet,

00:02:43.148 --> 00:02:45.077
we haven't figured out our B yet

00:02:45.077 --> 00:02:48.638
but our function is the natural log

00:02:48.638 --> 00:02:52.138
of X and then I will just write a DX here.

00:02:53.580 --> 00:02:55.199
So, in order to complete writing

00:02:55.199 --> 00:02:56.205
this definite integral

00:02:56.205 --> 00:02:58.887
I need to be able to write the upper bound

00:02:58.887 --> 00:03:00.553
and the way to figure out the upper bound

00:03:00.553 --> 00:03:03.189
is by looking at our delta X

00:03:03.189 --> 00:03:05.943
because the way that we
would figure out a delta X

00:03:05.943 --> 00:03:08.481
for this Riemann sum here,

00:03:08.481 --> 00:03:12.178
we would say that delta X
is equal to the difference

00:03:12.178 --> 00:03:15.304
between our bounds divided
by how many sections

00:03:15.304 --> 00:03:17.614
we want to divide it in, divided by N.

00:03:17.614 --> 00:03:20.031
So, it's equals to B minus A,

00:03:21.891 --> 00:03:23.891
B minus A over N, over N

00:03:29.404 --> 00:03:31.102
and so, you can pattern match here.

00:03:31.102 --> 00:03:34.391
If this is delta X is
equal to B minus A over N.

00:03:34.391 --> 00:03:35.520
Let me write this down.

00:03:35.520 --> 00:03:38.437
So, this is going to be equal to B,

00:03:39.364 --> 00:03:41.614
B minus our A which is two,

00:03:43.137 --> 00:03:44.720
all of that over N,

00:03:45.845 --> 00:03:47.012
so B minus two

00:03:50.510 --> 00:03:52.523
is equal to five

00:03:52.523 --> 00:03:55.754
which would make B equal to seven.

00:03:55.754 --> 00:03:57.861
B is equal to seven.

00:03:57.861 --> 00:03:58.699
So, there you have it.

00:03:58.699 --> 00:04:02.449
We have our original
limit, our Riemann limit

00:04:03.811 --> 00:04:05.551
or our limit of our Riemann sum

00:04:05.551 --> 00:04:08.654
being rewritten as a definite integral.

00:04:08.654 --> 00:04:09.623
And once again, I want to emphasize

00:04:09.623 --> 00:04:11.205
why this makes sense.

00:04:11.205 --> 00:04:13.182
If we wanted to draw this

00:04:13.182 --> 00:04:14.582
it would look something like this,

00:04:14.582 --> 00:04:19.356
I'm gonna try to hand draw
the natural log function,

00:04:19.356 --> 00:04:21.689
it looks something like this

00:04:27.386 --> 00:04:30.258
and this right over here would be one

00:04:30.258 --> 00:04:33.211
and so, let's say this is two

00:04:33.211 --> 00:04:35.664
and so going from two to seven,

00:04:35.664 --> 00:04:37.664
this isn't exactly right

00:04:38.582 --> 00:04:42.932
and so, our definite integral
is concerned with the area

00:04:42.932 --> 00:04:46.571
under the curve from two until seven

00:04:46.571 --> 00:04:48.154
and so, this Riemann sum you can view

00:04:48.154 --> 00:04:51.505
as an approximation when N
isn't approaching infinity

00:04:51.505 --> 00:04:52.538
but what you're saying is look,

00:04:52.538 --> 00:04:55.085
when I is equal to one,

00:04:55.085 --> 00:04:59.054
your first one is going to
be of width five over N,

00:04:59.054 --> 00:05:01.573
so this is essentially
saying our difference

00:05:01.573 --> 00:05:02.654
between two and seven,

00:05:02.654 --> 00:05:04.273
we're taking that distance five,

00:05:04.273 --> 00:05:06.319
dividing it into N rectangles,

00:05:06.319 --> 00:05:11.104
and so, this first one is
going to have a width of five

00:05:11.104 --> 00:05:14.172
over N and then what's
the height gonna be?

00:05:14.172 --> 00:05:16.270
Well, it's a right Riemann sum,

00:05:16.270 --> 00:05:19.966
so we're using the value of
the function right over there,

00:05:19.966 --> 00:05:22.298
write it two plus five over N.

00:05:22.298 --> 00:05:24.638
So, this value right over here.

00:05:24.638 --> 00:05:26.788
This is the natural log,

00:05:26.788 --> 00:05:30.121
the natural log of two plus five over N,

00:05:32.158 --> 00:05:33.996
and since this is the first rectangle

00:05:33.996 --> 00:05:35.746
times one, times one.

00:05:36.687 --> 00:05:38.564
Now we could keep going.

00:05:38.564 --> 00:05:40.310
This one right over here

00:05:40.310 --> 00:05:43.320
the width is the same, five over N

00:05:43.320 --> 00:05:45.218
but what's the height?

00:05:45.218 --> 00:05:47.988
Well, the height here,
this height right over here

00:05:47.988 --> 00:05:49.561
is going to be the natural log

00:05:49.561 --> 00:05:53.311
of two plus five over
N times two, times two.

00:05:55.150 --> 00:05:57.650
This is for I is equal to two.

00:05:58.484 --> 00:06:00.800
This is I is equal to one.

00:06:00.800 --> 00:06:02.725
And so, hopefully you are
seeing that this makes sense.

00:06:02.725 --> 00:06:04.879
The area of this first rectangle

00:06:04.879 --> 00:06:06.866
is going to be natural log of two

00:06:06.866 --> 00:06:09.038
plus five over N times one

00:06:09.038 --> 00:06:10.455
times five over N

00:06:12.201 --> 00:06:13.555
and the second one over here,

00:06:13.555 --> 00:06:17.305
natural log of two plus
five over N times two

00:06:18.905 --> 00:06:20.322
times five over N

00:06:21.498 --> 00:06:23.414
and so, this is calculating the sum

00:06:23.414 --> 00:06:25.384
of the areas of these rectangles

00:06:25.384 --> 00:06:28.263
but then it's taking the
limit as N approaches infinity

00:06:28.263 --> 00:06:30.046
so we get better and better approximations

00:06:30.046 --> 00:06:33.129
going all the way to the exact area.


Worked example: Rewriting definite integral as limit of Riemann sum

WEBVTT
Kind: captions
Language: en

00:00:02.962 --> 00:00:04.584
- [Instructor] Let's get some practice

00:00:04.584 --> 00:00:06.423
rewriting definite integrals

00:00:06.423 --> 00:00:08.882
as the limit of a Riemann sum.

00:00:08.882 --> 00:00:10.954
So let's say I wanted to
take the definite integral

00:00:10.954 --> 00:00:14.204
from pi to two pi of cosine of x dx.

00:00:18.234 --> 00:00:19.786
And I what I wanna do is I wanna write it

00:00:19.786 --> 00:00:24.533
as the limit as n approaches
infinity of a Riemann sum.

00:00:24.533 --> 00:00:27.037
So it's gonna take the form of the limit

00:00:27.037 --> 00:00:30.230
as n approaches infinity, and we can have

00:00:30.230 --> 00:00:32.435
our Sigma notation right over here,

00:00:32.435 --> 00:00:35.102
and I would say from, let's say

00:00:36.584 --> 00:00:39.755
i is equal to one all the way to n.

00:00:39.755 --> 00:00:41.071
Let me scroll down a little bit

00:00:41.071 --> 00:00:43.233
so it doesn't get all
squashed up at the top.

00:00:43.233 --> 00:00:45.989
Of, and so let's me draw
what's actually going on

00:00:45.989 --> 00:00:47.488
so that we can get a better sense

00:00:47.488 --> 00:00:51.136
of what to write here
within the Sigma notation.

00:00:51.136 --> 00:00:52.969
So let me do it large.

00:00:54.479 --> 00:00:57.229
So if this is pi right over here,

00:00:59.141 --> 00:01:01.724
this would be three pi over two

00:01:03.744 --> 00:01:07.827
and this would be two pi
right over here, two pi.

00:01:10.511 --> 00:01:13.570
Now what does the graph of cosine of x do?

00:01:13.570 --> 00:01:16.670
Well, at pi cosine of pi is negative one.

00:01:16.670 --> 00:01:18.141
So that's negative one there.

00:01:18.141 --> 00:01:20.474
And cosine of two pi is one.

00:01:21.470 --> 00:01:23.575
And so the graph is gonna
do something like this,

00:01:23.575 --> 00:01:26.743
and this is obviously just
a hand-drawn version of it,

00:01:26.743 --> 00:01:29.689
but you have seen cosine functions before,

00:01:29.689 --> 00:01:31.420
this is just part of it.

00:01:31.420 --> 00:01:33.379
And so this definite integral represents

00:01:33.379 --> 00:01:35.546
the area from pi to two pi

00:01:36.825 --> 00:01:40.251
between the curve and the x axis.

00:01:40.251 --> 00:01:43.301
And you might already know
that this area is going to be,

00:01:43.301 --> 00:01:45.800
or this part of the definite
integral'd be negative

00:01:45.800 --> 00:01:47.991
and this would be positive,
and it will cancel out

00:01:47.991 --> 00:01:51.663
and this would all actually
end up being zero in this case.

00:01:51.663 --> 00:01:55.853
But this exercise for this
video is to rewrite this

00:01:55.853 --> 00:02:00.408
as a limit as n approaches
infinity of a Riemann sum.

00:02:00.408 --> 00:02:03.000
So there's a Riemann sum, what we wanna do

00:02:03.000 --> 00:02:04.730
is think about breaking this up

00:02:04.730 --> 00:02:07.429
into a bunch of rectangles.

00:02:07.429 --> 00:02:10.567
So let's say, or I
should say n rectangles.

00:02:10.567 --> 00:02:14.681
So that's our first one right over there,

00:02:14.681 --> 00:02:16.504
then this might be our second one,

00:02:16.504 --> 00:02:18.420
and let's do right Riemann sum

00:02:18.420 --> 00:02:21.728
where the right boundary of our rectangle,

00:02:21.728 --> 00:02:24.012
what the value of the
function is at that point,

00:02:24.012 --> 00:02:25.805
that's what defines the height.

00:02:25.805 --> 00:02:29.529
So that's our second one
all the way until this one

00:02:29.529 --> 00:02:32.282
right over here is going
to be our n-th one.

00:02:32.282 --> 00:02:34.951
So this is one, let me write it this way.

00:02:34.951 --> 00:02:36.447
This is i is equal to one,

00:02:36.447 --> 00:02:39.188
this is i is equal to two, all the way

00:02:39.188 --> 00:02:41.855
until we get to i is equal to n.

00:02:42.962 --> 00:02:45.792
And then if we take the limit
as n approaches infinity,

00:02:45.792 --> 00:02:47.700
the sum of the areas of these rectangles

00:02:47.700 --> 00:02:49.892
are gonna get better
and better and better.

00:02:49.892 --> 00:02:52.141
And so let's first think
about it, what is the width

00:02:52.141 --> 00:02:55.286
of each of these rectangles going to be?

00:02:55.286 --> 00:02:59.035
Well, I am taking this
interval from pi to two pi

00:02:59.035 --> 00:03:02.006
and I'm gonna divide it
into n equal intervals.

00:03:02.006 --> 00:03:04.492
So the width of each of these,

00:03:04.492 --> 00:03:06.767
the width of each of these is going to be

00:03:06.767 --> 00:03:09.766
two pi minus pi, so I'm just taking

00:03:09.766 --> 00:03:12.267
the difference between
my bounds of integration,

00:03:12.267 --> 00:03:16.434
and I am dividing by n,
which is equal to pi over n.

00:03:18.501 --> 00:03:19.845
So that's the width of each of these.

00:03:19.845 --> 00:03:22.839
That's pi over n, this is
pi over n, this is pi of n.

00:03:22.839 --> 00:03:26.447
And what's the height of
each of these rectangles?

00:03:26.447 --> 00:03:28.476
Remember this is a right Riemann sum.

00:03:28.476 --> 00:03:33.402
So it's going to be the
right end of our rectangle

00:03:33.402 --> 00:03:34.996
is going to define the height.

00:03:34.996 --> 00:03:37.600
So this right over here,
what would this height be?

00:03:37.600 --> 00:03:41.722
Well, this height, this
value I should say,

00:03:41.722 --> 00:03:44.972
this is going to be equal to f of what?

00:03:46.509 --> 00:03:49.209
Well, this was pi and
this is going to be pi

00:03:49.209 --> 00:03:52.917
plus the length of our
interval right over here,

00:03:52.917 --> 00:03:54.613
the base of the rectangle.

00:03:54.613 --> 00:03:58.456
So we started at pi, so
it's gonna be pi plus,

00:03:58.456 --> 00:04:02.623
this was going to be pi over
n, I could say times one.

00:04:04.031 --> 00:04:06.521
That's this height right over here.

00:04:06.521 --> 00:04:09.273
What's this one going
to be right over here?

00:04:09.273 --> 00:04:12.440
Well, this one is going to be f of pi,

00:04:13.427 --> 00:04:17.010
our first start, plus
pi over n times what?

00:04:18.310 --> 00:04:20.979
We're gonna have pi over n two times.

00:04:20.979 --> 00:04:22.646
Pi over n times two.

00:04:23.788 --> 00:04:26.981
So the general form of the right boundary

00:04:26.981 --> 00:04:29.542
is going to be, so for
example, this height

00:04:29.542 --> 00:04:32.853
right over here, this is going to be f of,

00:04:32.853 --> 00:04:34.686
we started at pi plus,

00:04:35.769 --> 00:04:37.932
we're doing the right Riemann sum,

00:04:37.932 --> 00:04:42.099
so we're gonna add pi over
n n times by this point.

00:04:43.308 --> 00:04:44.808
Pi over n times n.

00:04:48.111 --> 00:04:50.323
Or, if we wanted to say it generally,

00:04:50.323 --> 00:04:52.242
if we're talking about the i-th rectangle,

00:04:52.242 --> 00:04:55.133
remember we sum them all
up, what's the height?

00:04:55.133 --> 00:04:57.436
Well, the height is
going to be, in this case

00:04:57.436 --> 00:05:00.353
it's going to be cosine of pi plus,

00:05:04.696 --> 00:05:07.337
if we're with the i-th rectangle,

00:05:07.337 --> 00:05:10.846
we are going to add pi over n i times.

00:05:10.846 --> 00:05:12.346
Pi over n times i.

00:05:14.310 --> 00:05:17.368
And then, that's the height
of each of our rectangles.

00:05:17.368 --> 00:05:18.238
And then what's the width?

00:05:18.238 --> 00:05:20.028
Well, we already figured that out.

00:05:20.028 --> 00:05:21.361
Times pi over n.

00:05:22.319 --> 00:05:23.892
And if you wanna be careful and make sure

00:05:23.892 --> 00:05:26.217
that this Sigma notation
applies to the whole thing,

00:05:26.217 --> 00:05:27.441
there you have it.

00:05:27.441 --> 00:05:29.936
We have just re-expressed
this definite integral

00:05:29.936 --> 00:05:32.936
as the limit of a right Riemann sum.


Worked examples: Summation notation

WEBVTT
Kind: captions
Language: en

00:00:00.211 --> 00:00:02.294
- [Instructor] We're told consider the sum

00:00:02.294 --> 00:00:05.399
two plus five plus eight plus 11.

00:00:05.399 --> 00:00:09.046
Which expression is
equal to the sum above?

00:00:09.046 --> 00:00:12.435
And they tell us choose
all answers that apply.

00:00:12.435 --> 00:00:13.943
So like always, pause the video,

00:00:13.943 --> 00:00:16.814
and see if you can work
through this on your own.

00:00:16.814 --> 00:00:18.178
So when you look at the sum,

00:00:18.178 --> 00:00:20.037
it's clear you're starting at two.

00:00:20.037 --> 00:00:23.148
You're adding three each time.

00:00:23.148 --> 00:00:25.975
And we also are dealing with

00:00:25.975 --> 00:00:28.401
you have four total terms.

00:00:28.401 --> 00:00:30.605
Now we could try to
construct an equation here

00:00:30.605 --> 00:00:32.909
or an expression using sigma notation.

00:00:32.909 --> 00:00:35.229
But instead, what I like to
do, is look at our options.

00:00:35.229 --> 00:00:37.783
We really only have to look
at these two options here

00:00:37.783 --> 00:00:39.226
and expand them out.

00:00:39.226 --> 00:00:41.873
What sum would each of these be?

00:00:41.873 --> 00:00:43.673
Well, this is going to be the same thing

00:00:43.673 --> 00:00:45.649
as we're starting at n equals one.

00:00:45.649 --> 00:00:48.714
So this is going to be three
times, when n equals one,

00:00:48.714 --> 00:00:50.745
three times one minus one.

00:00:50.745 --> 00:00:54.035
And then plus, then
we'll go to n equals two.

00:00:54.035 --> 00:00:56.202
Three times two minus one.

00:00:57.135 --> 00:00:59.262
Then we're gonna go to n equals three.

00:00:59.262 --> 00:01:02.262
So plus three times three minus one.

00:01:03.486 --> 00:01:06.138
And then finally, we're
gonna go to n equals four.

00:01:06.138 --> 00:01:08.777
So plus three times, when n equals four,

00:01:08.777 --> 00:01:12.002
this is three times four minus one.

00:01:12.002 --> 00:01:16.126
So just to be clear, this is
what we did when n equals one.

00:01:16.126 --> 00:01:16.959
Let me write it down.

00:01:16.959 --> 00:01:18.287
N equals one.

00:01:18.287 --> 00:01:22.170
This is what we got when n equals two.

00:01:22.170 --> 00:01:26.002
This is what we got when n equals three.

00:01:26.002 --> 00:01:30.002
And this is what we got
when n is equal to four.

00:01:31.434 --> 00:01:33.667
And we stopped at n equals four

00:01:33.667 --> 00:01:35.692
'cause it tells us right over there,

00:01:35.692 --> 00:01:36.663
we start at n equals one

00:01:36.663 --> 00:01:38.568
and we go all of the way to n equals four.

00:01:38.568 --> 00:01:40.218
So what does this equal?

00:01:40.218 --> 00:01:44.072
So let's see, three times
one minus one is two.

00:01:44.072 --> 00:01:45.978
So this is looking good so far.

00:01:45.978 --> 00:01:48.587
Three times two minus one,

00:01:48.587 --> 00:01:49.938
that's six minus one.

00:01:49.938 --> 00:01:51.279
That is five.

00:01:51.279 --> 00:01:52.810
So still looking good.

00:01:52.810 --> 00:01:56.047
Three times three minus
one, that's nine minus one.

00:01:56.047 --> 00:01:58.059
Once again, still looking good.

00:01:58.059 --> 00:02:00.487
Three times four minus one,

00:02:00.487 --> 00:02:01.404
that is 11.

00:02:02.365 --> 00:02:03.629
So we like this choice,

00:02:03.629 --> 00:02:05.714
so I would definitely select this one.

00:02:05.714 --> 00:02:08.197
Now let's do the same thing over here.

00:02:08.197 --> 00:02:10.197
When n is equal to zero,

00:02:11.345 --> 00:02:15.379
it's going to be two
plus three times zero.

00:02:15.379 --> 00:02:17.177
So that's just going to be two.

00:02:17.177 --> 00:02:20.107
And then plus, when n is equal to one,

00:02:20.107 --> 00:02:23.059
it's going to be two plus three times one,

00:02:23.059 --> 00:02:24.032
which is five.

00:02:24.032 --> 00:02:25.474
This is starting to look good.

00:02:25.474 --> 00:02:28.028
Now when n is equal to,
this was n equals zero,

00:02:28.028 --> 00:02:29.129
this was n equals one,

00:02:29.129 --> 00:02:30.925
so now we're at n equals two.

00:02:30.925 --> 00:02:32.367
So at n equals two,

00:02:32.367 --> 00:02:34.367
two plus three times two

00:02:35.570 --> 00:02:38.503
is two plus six, which is eight.

00:02:38.503 --> 00:02:39.336
And this makes sense.

00:02:39.336 --> 00:02:42.511
Every time we increase n by one,

00:02:42.511 --> 00:02:45.135
we are adding another three,

00:02:45.135 --> 00:02:46.606
which is consistent
with what we saw there.

00:02:46.606 --> 00:02:48.702
We start at two, when n equals zero,

00:02:48.702 --> 00:02:50.822
this is just, that three n is just zero.

00:02:50.822 --> 00:02:52.556
So you start at two, and then you keep,

00:02:52.556 --> 00:02:55.094
every time you increase n by one,

00:02:55.094 --> 00:02:56.943
you are just adding three again.

00:02:56.943 --> 00:02:58.727
So finally, when n is equal to three,

00:02:58.727 --> 00:03:02.370
two plus three times three is two plus,

00:03:02.370 --> 00:03:03.845
or is

00:03:03.845 --> 00:03:05.538
11, I should say.

00:03:05.538 --> 00:03:08.731
And so this also is exactly the same sum.

00:03:08.731 --> 00:03:12.137
So I feel good about both of these.

00:03:12.137 --> 00:03:14.304
Let's do one more example.

00:03:15.962 --> 00:03:18.102
So we're here, we're given the sum,

00:03:18.102 --> 00:03:19.711
and we're saying choose one answer.

00:03:19.711 --> 00:03:22.881
Which of these is equivalent
to this sum right over here?

00:03:22.881 --> 00:03:25.556
Well, like we did before,
let's just expand it out.

00:03:25.556 --> 00:03:28.562
And what's different here is
that we just have a variable,

00:03:28.562 --> 00:03:31.430
but that shouldn't make it
too much more difficult.

00:03:31.430 --> 00:03:33.299
So let's do the situation,
and I'll write it out.

00:03:33.299 --> 00:03:35.901
Let's do the situation
when n is equal to one.

00:03:35.901 --> 00:03:40.823
When n is equal to one, it's
gonna be k over one plus one.

00:03:40.823 --> 00:03:43.897
K over one plus one,
and I'll write it out.

00:03:43.897 --> 00:03:45.894
This is n is equal to one.

00:03:45.894 --> 00:03:48.469
And then plus, when n is equal to two,

00:03:48.469 --> 00:03:50.206
it's gonna be k

00:03:50.206 --> 00:03:51.706
over two plus one.

00:03:52.695 --> 00:03:55.113
This is n is equal to two.

00:03:55.113 --> 00:03:56.240
And then we're gonna keep going.

00:03:56.240 --> 00:03:57.741
When n is equal to three,

00:03:57.741 --> 00:03:59.544
it's gonna be k

00:03:59.544 --> 00:04:01.211
over three plus one.

00:04:02.317 --> 00:04:04.459
That's n is equal to three.

00:04:04.459 --> 00:04:05.987
And then plus, finally,

00:04:05.987 --> 00:04:09.151
'cause we stop right over
here at n equals four,

00:04:09.151 --> 00:04:13.318
when n equals four, it's going
to be k over four plus one.

00:04:14.968 --> 00:04:17.150
That's when n is equal to four.

00:04:17.150 --> 00:04:18.843
So this is all going to be equal,

00:04:18.843 --> 00:04:19.910
I'll just write it over here,

00:04:19.910 --> 00:04:22.935
this is equal to k over two,

00:04:22.935 --> 00:04:24.435
plus k over three,

00:04:25.586 --> 00:04:27.003
plus k over four,

00:04:28.743 --> 00:04:30.970
plus k over five,

00:04:30.970 --> 00:04:34.303
which is exactly this choice right over,

00:04:35.930 --> 00:04:37.395
right over here.

00:04:37.395 --> 00:04:39.600
And actually, if I had looked
at the choices ahead of time,

00:04:39.600 --> 00:04:41.512
I might have even been
able to save even more time

00:04:41.512 --> 00:04:43.832
by just saying, well, look,

00:04:43.832 --> 00:04:45.939
actually, if you just try
to compute the first term,

00:04:45.939 --> 00:04:48.655
when n is equal to one,
it would be k over two.

00:04:48.655 --> 00:04:52.140
Well, only this one is
starting with a k over two.

00:04:52.140 --> 00:04:55.695
This, this one has no k's
here, which is sketchy.

00:04:55.695 --> 00:04:57.137
They're trying to look at the error

00:04:57.137 --> 00:05:00.422
where you try to replace the
k with the number as well,

00:05:00.422 --> 00:05:01.361
not just the n.

00:05:01.361 --> 00:05:03.017
So that's what they're trying to do here.

00:05:03.017 --> 00:05:05.719
Here they're trying to, let's see,

00:05:05.719 --> 00:05:10.091
well, this isn't as obvious
what they're even trying to do,

00:05:10.091 --> 00:05:12.619
right over here where they
put the k in the denominator.

00:05:12.619 --> 00:05:15.369
And here if you swapped the n and the k's,

00:05:15.369 --> 00:05:17.682
then you would've gotten
this thing right over here.

00:05:17.682 --> 00:05:20.524
So we definitely feel good about choice A.