Use polar coordinates to find the volume of the given solid.

[ryantc]

Homework Statement

1. Use polar coordinates to find the volume of the given solid.
2. Inside the sphere x^2 + y^2 + z^2 = 16 and outside the cylinder x^2 + y^2 = 4.

2. The attempt at a solution

My attempt as following:
2<=r<=4, and 0<=theta<=2pi

So I do a double integral of f(x,y)=sqrt(16-r^2)r dr d(theta) and it gives me 16sqrt(3)pi

but I saw the answer somewhere, the volume is 2 times the answer I got, I cannot figure out why 2 times? and in the answer it says "by symmetry", any explanation will be appreciated.

 

[xaos]

you found the area of the annulus in the polar plane (two integrals: rdrdt), but you need to set your upper and lower bounds wrt the upper and lower halves of the sphere(third integral dz). the double negatives sum to twice your value.

[edit] a little backwards, dz is being evaluated first.

 

[ryantc]

Thanks, I think I got the idea here, when I set z=0 I get the sphere on xy plane together with the cylinder to find the range of r,

But I just forget the lower half, that's where the 2 times comes from . :(