Triple integral problem (Multivariable calculus)

[Aynrand12]

Hey,

I'm stuck on this problem: (The sphere has a radius of 2)


I understand the limits in spherical coordinates to be:
0< r < 2
0 < θ < 2pi
0 < φ < pi

I'm having trouble understanding what to do with the xy in spherical coordinates.
I know:
x^2 + y^2 = r^2sin^2φ
dxdydz =r^2sinφι

But xy = r^2sin^2 * cosθsinθ and i'm having a hard time simplyfing this expression into something that makes it possible for me to integrate..

Would be grateful for some hints on what i could do to solve this.
Thanks

 

[LoffieZA]

Hey,

Please see the attached images. I am hoping it clears things up for you. Also take note that you need to multiply by P^2 sin(phi) when converting from cartesian to spherical coordinates. That is where the p^2 sin(phi) comes from at the end of the line.

 

[Aynrand12]

Hey,

Please see the attached images. I am hoping it clears things up for you. Also take note that you need to multiply by P^2 sin(phi) when converting from cartesian to spherical coordinates. That is where the p^2 sin(phi) comes from at the end of the line.

Hi!
yes precisely, that's what i have done now as well.
Would an appropriate way to continue solving the integral be to split the integral into two integrals and compute seperately (Law of addition) or how would you recommend computing it?

Thank you!