Question

Find the volume between two intersecting cylinders x^2+y^2=r^2 and y^2+z^2=r^2 using polar coordinates and double integrals only.

Answer 1

in the xy-plane\[-\sqrt{r^2-y^2}\le x\le\sqrt{r^2-y^2}\]in the yz-plane\[-\sqrt{r^2-y^2}\le z\le\sqrt{r^2-y^2}\]leaving\[-r\le y\le r\]hm...

Answer 2

in the xz-plane the intersection is square with sides=2r

Answer 3

I think the integral is\[\iint\limits_Dr\sin\theta dA=\int_0^{2\pi}\int_0^r r^2\sin\theta drd\theta\]can you check somewhow?

Answer 4

actually it would have to be\[\iint\limits_Dr\sin\theta dA=2\int_0^\pi\int_0^r r^2\sin\theta drd\theta\]to avoid getting zero

Answer 5

But the integrand should be r^2 *
cos(theta) ?

Answer 6

since i am integrating z = sqrt(z^2-y^2)

Answer 7

so in polar i thought it was sqrt(r^2-y^2) = x = rcos(theta)

Answer 8

yeah I just came to that conclusion as well

Answer 9

I guess I was right the first time

Answer 10

but if I integrate with rcos(theta), I end up with zero in the end

Answer 11

I will get sin(2*pi) - sin(0) = 0.

Answer 12

Hm.. now I am not sure again. No way to check I presume...

Answer 13

@mahmit2012 double integral help