# Question c1f8f

Sep 8, 2017

Volume of bounded area $= \textcolor{red}{9 \pi}$ cubic units

#### Explanation:

First let's consider this within the XY-plane
$y = \sqrt{9 - {x}^{2}}$
is the equation for the top half (since the root symbol restricts us to non-negative vales) of a circle with center the origin and radius $\sqrt{9} = 3$
bounding this by $y = 0$ and $x = 0$
implies we are dealing (within the XY-plane) with a quarter circle (whose radius is $3$ units)

If we rotate this about the X-axis we will obtain half of a sphere.

The volume of a sphere is $V = \frac{4}{3} \pi {r}^{3}$

Here the radius, $r = 3$, and we only want half of the sphere.

So
color(white)("XXX")"Volume"_"bound region"=1/2 xx 4/3pi xx3^3#

$\textcolor{w h i t e}{\text{XXX}} = 9 \pi$ (cubic units)

I have solved this way: