# How do you find the volume of the solid formed by rotating the region enclosed by ?

Apr 4, 2016

$V = \pi \left({e}^{0.6} / 2 + 4 {e}^{0.3} - 3.3\right) \approx 9.4577 \ldots$

#### Explanation:

We'll use a method call the disk method to do this.

Take a look at the above picture. Pretend it's an infinitely small cylindrical disk, with radius $r$ and height $\mathrm{dx}$. When we revolve $y = {e}^{x} + 2$ about the $x$-axis, we can break the resulting solid down to infinitely many infinitely small cylindrical disks like the one above (woah, calculus huh!). All of these disks have a radius equal to the value of the $y$ and height equal to an infinitely small change in $x$, or $\mathrm{dx}$.

We know volume of a cylinder is given by:
$V = \pi {r}^{2} h$
And we know that the radius of each cylinder is the value of $y$, and the height is $\mathrm{dx}$; that means the volume of an infinitely small disk is:
$\mathrm{dV} = \pi \left({y}^{2}\right) \mathrm{dx}$
Integrating both sides to find total volume, for $x = 0$ to $x = 0.3$, gives
$V = {\int}_{0}^{0.3} \pi \left({y}^{2}\right) \mathrm{dx}$
But $\pi$ is a constant and $y = {e}^{x} + 2$, so
$V = \pi {\int}_{0}^{0.3} {\left({e}^{x} + 2\right)}^{2} \mathrm{dx}$

Expanding the ${\left({e}^{x} + 2\right)}^{2}$ binomial:
$V = \pi {\int}_{0}^{0.3} {e}^{2 x} + 4 {e}^{x} + 4 \mathrm{dx}$
And using the sum rule:
$V = \pi {\int}_{0}^{0.3} {e}^{2 x} \mathrm{dx} + \pi {\int}_{0}^{0.3} 4 {e}^{x} \mathrm{dx} + \pi {\int}_{0}^{0.3} 4 \mathrm{dx}$
Finally, evaluating these one by one:
$V = \pi {\left[{e}^{2 x} / 2\right]}_{0}^{0.3} + 4 \pi {\left[{e}^{x}\right]}_{0}^{0.3} + 4 \pi {\left[x\right]}_{0}^{0.3}$
$V = \pi \left(\left({e}^{0.6} / 2 - {e}^{0} / 2\right) + 4 \left({e}^{0.3} - {e}^{0}\right) + 4 \left(0.3 - 0\right)\right)$
$V = \pi \left({e}^{0.6} / 2 - \frac{1}{2} + 4 {e}^{0.3} - 4 + 1.2\right)$
$V = \pi \left({e}^{0.6} / 2 + 4 {e}^{0.3} - 3.3\right) \approx 9.4577 \ldots$

Apr 11, 2016

Awesome Ken...

Here is a Maple rendering of the rotation!

By the way, you have the coolest name in the world.

May 12, 2016

0.9117

#### Explanation:

$f \left(r\right) = {e}^{x}$
$V \left(R\right) = 2 \pi {\int}_{0}^{R} f \left(r\right) r \mathrm{dr}$
$C \left(h , r\right) = \pi h {r}^{2}$ now with $h = 2$ and $R = 0.3$
${V}_{t} = V \left(R\right) + C \left(2 , R\right) = 0.3462 + 0.5655 = 0.9117$