Question

Question #c1f8f

Answer 1

Volume of bounded area `=color(red)(9pi)` 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=4/3pir^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`

`color(white)("XXX")=9pi` (cubic units)

Answer 2

I have solved this way:

Explanation: