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## Homework Statement

Bounded by the cylinders x

^{2}+ y

^{2}= r

^{2}and y

^{2}+ z

^{2}= r

^{2}

We're supposed to stick to double integrals as triple integrals are taught in a later section.

## The Attempt at a Solution

Edit:

Alright, I think I go to the right answer.

x = sqrt(r

^{2}- y

^{2})

z = sqrt(r[SUP2[/SUP] - y

^{2})

The bounds are 0 < y < r and 0 < x < sqrt(r

^{2}- y

^{2})

So I get:

integrate integrate sqrt(r

^{2}- y

^{2}) dx from 0 to sqrt(r

^{2}- y

^{2}) dy from 0 to r

The first integral goes to:

sqrt(r

^{2}- y

^{2}) * x from 0 to sqrt(r

^{2}- y

^{2}) =

(r

^{2}- y

^{2})

The second integral then makes it:

r

^{2}* y - 1/3 * y

^{3}from 0 to r which gives us:

2/3 * r

^{3}

However, you're supposed to throw in an 8 at the beginning to times everything by (why?) so you get:

16/3 * r

^{3}

So, my questions:

Was my edited solution correct and why are you supposed to multiple the entire double integral by 8.

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