## math1234 2 years ago 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.

1. TuringTest

|dw:1352144686186:dw|

2. TuringTest

in the xy-plane|dw:1352144919394:dw|$-\sqrt{r^2-y^2}\le x\le\sqrt{r^2-y^2}$in the yz-plane|dw:1352145089843:dw|$-\sqrt{r^2-y^2}\le z\le\sqrt{r^2-y^2}$leaving$-r\le y\le r$hm...

3. TuringTest

in the xz-plane the intersection is square with sides=2r|dw:1352145471598:dw|

4. TuringTest

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?

5. TuringTest

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

6. math1234

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

7. math1234

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

8. math1234

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

9. TuringTest

yeah I just came to that conclusion as well

10. TuringTest

I guess I was right the first time

11. math1234

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

12. math1234

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

13. TuringTest

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

14. TuringTest

@mahmit2012 double integral help