Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r2
We're supposed to stick to double integrals as triple integrals are taught in a later section.
The Attempt at a Solution
Alright, I think I go to the right answer.
x = sqrt(r2 - y2)
z = sqrt(r[SUP2[/SUP] - y2)
The bounds are 0 < y < r and 0 < x < sqrt(r2 - y2)
So I get:
integrate integrate sqrt(r2 - y2) dx from 0 to sqrt(r2 - y2) dy from 0 to r
The first integral goes to:
sqrt(r2 - y2) * x from 0 to sqrt(r2 - y2) =
(r2 - y2)
The second integral then makes it:
r2 * y - 1/3 * y3 from 0 to r which gives us:
2/3 * r3
However, you're supposed to throw in an 8 at the beginning to times everything by (why?) so you get:
16/3 * r3
So, my questions:
Was my edited solution correct and why are you supposed to multiple the entire double integral by 8.