The area under the curve y=e^-x between x=0 and x=1 is rotated about the x axis find the volume?

Apr 10, 2018

Volume is $\frac{\pi}{2} \left(1 - {e}^{-} 2\right) = 1.358$ cubic units.

Explanation:

Let us see the graph of $y = {e}^{- x}$ between $x = 0$ and $x = 1$.

graph{e^(-x) [-2.083, 2.917, -0.85, 1.65]}

To find the desired volume the shaded portion (shown below, will have to be rotated around $x$-axis.

As volume of a cylinder is $\pi {r}^{2} h$, here we will have $r = {e}^{- x}$ and $h = \mathrm{dx}$

and hence volume would be

${\int}_{0}^{1} \pi {e}^{- 2 x} \mathrm{dx}$

= $\pi {\int}_{0}^{1} {e}^{- 2 x} \mathrm{dx}$

= $\pi \times {\left[- {e}^{- 2 x} / 2\right]}_{0}^{1}$

= $\pi \left[- {e}^{- 2} / 2 + \frac{1}{2}\right]$

= $\frac{\pi}{2} \left(1 - {e}^{-} 2\right)$

= $\frac{\pi}{2} \left(1 - 0.1353\right)$

= $1.358$