```python
from IPython.display import Image
from IPython.core.display import HTML 
from sympy import *
Image(url= "https://i.imgur.com/wo5OwkX.png")
```









```python
f = Function("f")
x,y = symbols("x y")
e0 = Integral(f(x),(x,-pi,pi))
e0
```




$\displaystyle \int\limits_{- \pi}^{\pi} f{\left(x \right)}\, dx$




```python
a  = 2*x**3
b = 5*sin(x)
e3 = Piecewise((a, ((x < 0)) & ((x >= -pi))),(b,((x<=pi))& ((x >= 0))))
e4 = Eq(f(x),e3)
e4

```




$\displaystyle f{\left(x \right)} = \begin{cases} 2 x^{3} & \text{for}\: x \geq - \pi \wedge x < 0 \\5 \sin{\left(x \right)} & \text{for}\: x \geq 0 \wedge x \leq \pi \end{cases}$




```python
e5 = Integral(e4,(x,-pi,pi))
e5
```




$\displaystyle \int\limits_{- \pi}^{\pi} f{\left(x \right)}\, dx = \int\limits_{- \pi}^{\pi} \begin{cases} 2 x^{3} & \text{for}\: x \geq - \pi \wedge x < 0 \\5 \sin{\left(x \right)} & \text{for}\: x \geq 0 \wedge x \leq \pi \end{cases}\, dx$




```python
e5.doit()
```




$\displaystyle \int\limits_{- \pi}^{\pi} f{\left(x \right)}\, dx = 10 - \frac{\pi^{4}}{2}$




```python
print(e5.doit().rhs)
```

    10 - pi**4/2



```python
Image(url= "https://i.imgur.com/XAC2vgG.png")
```









```python
Image(url= "https://i.imgur.com/z77oGr5.png")
```









```python
e6 = 1/(cos(x)**2)
e7 = Integral(e6,(x,-pi/4,0))
e7
```




$\displaystyle \int\limits_{- \frac{\pi}{4}}^{0} \frac{1}{\cos^{2}{\left(x \right)}}\, dx$




```python
e7.doit()
```




$\displaystyle 1$




```python
Image(url= "https://i.imgur.com/F7HIMOE.png")
```









```python
Image(url= "https://i.imgur.com/30ggSC2.png")
```









```python
e8 = x*(x+2)*(1-x)
e9 = Integral(e8,(x,-2,1))
e9
```




$\displaystyle \int\limits_{-2}^{1} x \left(1 - x\right) \left(x + 2\right)\, dx$




```python
e9.doit()
```




$\displaystyle - \frac{9}{4}$




```python
Image(url= "https://i.imgur.com/hyJJz6w.png")
```









```python
Image(url= "https://i.imgur.com/xUGdasC.png")
```









```python
e10 = abs(x**3)
e11 = Integral(e10,(x,-2,1))
e11
```




$\displaystyle \int\limits_{-2}^{1} \left|{x^{3}}\right|\, dx$




```python
e11.doit()
```




$\displaystyle \frac{17}{4}$




```python
Image(url= "https://i.imgur.com/rYiDqF4.png")
```









```python
Image(url= "https://i.imgur.com/vnZPXrc.png")
```









```python
t = symbols('t')
e12 = sqrt(t)*(5+t)
e13 = Integral(e12,(t,1,4))
e13
```




$\displaystyle \int\limits_{1}^{4} \sqrt{t} \left(t + 5\right)\, dt$




```python
e13.doit()
```




$\displaystyle \frac{536}{15}$




```python
Image(url= "https://i.imgur.com/RPOvgy8.png")
```









```python
Image(url= "https://i.imgur.com/vmAzFRJ.png")
```









```python
e14= sqrt(2/x)
e15 = Integral(e14,(x,1,4))
e15
```




$\displaystyle \int\limits_{1}^{4} \sqrt{2} \sqrt{\frac{1}{x}}\, dx$




```python
e15.doit()
```




$\displaystyle 2 \sqrt{2}$




```python
Image(url= "https://i.imgur.com/hcQlywf.png")
```









```python
Image(url= "https://i.imgur.com/iyZsZHs.png")
```









```python
e15= t**3 - 4*t**2
e16 = Integral(e15,(t,-1,5))
e16
```




$\displaystyle \int\limits_{-1}^{5} \left(t^{3} - 4 t^{2}\right)\, dt$




```python
e16.doit()
```




$\displaystyle -12$




```python
e17 = abs(e15)
e18 = Integral(e17,(t,-1,5))
e18
```




$\displaystyle \int\limits_{-1}^{5} \left|{t^{3} - 4 t^{2}}\right|\, dt$




```python
e18.doit()
```




$\displaystyle \frac{203}{6}$




```python
Image(url= "https://i.imgur.com/7uplphF.png")
```









```python
Image(url= "https://i.imgur.com/d9Fbxag.png")
```









```python
e20 = t**Rational(1,2)+3
e21 = Integral(e20,(t,5,9))
e21
```




$\displaystyle \int\limits_{5}^{9} \left(\sqrt{t} + 3\right)\, dt$




```python
a = e21.doit().evalf()
a
```




$\displaystyle 22.5464400750007$




```python
a/4
```




$\displaystyle 5.63661001875017$




```python
Image(url= "https://i.imgur.com/kG9utPJ.png")
```









```python
Image(url= "https://i.imgur.com/1jAeHgI.png")
```









```python
v = Function("v")
r,n,l,R,P = symbols('r n l R P')
v(r)
```




$\displaystyle v{\left(r \right)}$




```python
e22 = (P/(4*n*l)   )*(R**2-r**2)
e23 = Eq(v(r),e22)    
e23
```




$\displaystyle v{\left(r \right)} = \frac{P \left(R^{2} - r^{2}\right)}{4 l n}$




```python
e24 = Integral(e23.rhs,(r,0,R))
e24
```




$\displaystyle \int\limits_{0}^{R} \frac{P \left(R^{2} - r^{2}\right)}{4 l n}\, dr$




```python
e24.doit()/R
```




$\displaystyle \frac{P R^{2}}{6 l n}$




```python
print(e24.doit()/R)
```

    P*R**2/(6*l*n)



```python
e40 = P*R**2/(6*l*n)
e40
# source 06-05-018_Average_Value_of_a_Function.pdf

```




$\displaystyle \frac{P R^{2}}{6 l n}$




```python
Image(url= "https://i.imgur.com/3c17nUy.png")
```









```python
Image(url= "https://i.imgur.com/psTFm7P.png")
```









```python
e30 = x**2
e31 = Integral(e30,(x,3,6))
e31
```




$\displaystyle \int\limits_{3}^{6} x^{2}\, dx$




```python
e31.doit()
```




$\displaystyle 63$




```python
63/3
```




    21.0




```python
Image(url= "https://i.imgur.com/CeUw8Mi.png")
```









```python
Image(url= "https://i.imgur.com/af7s3HK.png")
```









```python
e32 = 0.2*x**2+130*x+200
e33 = Integral(e32,(x,0,450))
e33
```




$\displaystyle \int\limits_{0}^{450} \left(0.2 x^{2} + 130 x + 200\right)\, dx$




```python
e33.doit()/450
```




$\displaystyle 42950.0$




```python
Image(url= "https://i.imgur.com/xII0vO4.png")
```








1fa53a04a29ba80b2ff104ddd31e07e2ab4c702f
ipynb
Jupyter Notebook
Calculus_Homework/WWB19.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB19.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB19.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES