Volume of solid inside sphere and cylinder off by factor of 2

[stratusfactio]

Homework Statement

Find the volume V of the solid inside both x^2 + y^2 + z^2 = 4 and x^2 + y^2 = 1.

Homework Equations

So I get how to set it up; you use cylindrical coordinates because it makes life a whole lot simpler BUT the answer is (4pi/3)(8-3^(3/2)) and I got (2pi/3)(8-3^(3/2)). So as you can see, I'm off by a factor of 2.

The Attempt at a Solution

The integrands I have are: z=(0,sqrt(4-r^2); theta = (0, 2pi) and r = (0,1). Since I'm off by a factor of 2, I'm thinking that for theta I should integrate from 0 to 4pi, but conceptually I don't get why.

Any and all help would be much appreciated! :D

 

[LCKurtz]

You just found the volume above the xy plane. There is an equal volume below the xy plane.

 

[stratusfactio]

I love this freaking forum! so would theta range from -2pi to 2pi ? I only did with respect to the positive axis. Or even simpler just multiply by 2 because I found the area for just half of the xy plane?

 

[LCKurtz]

I love this freaking forum! so would theta range from -2pi to 2pi ? I only did with respect to the positive axis. Or even simpler just multiply by 2 because I found the area for just half of the xy plane?

No about the theta; that isn't where the problem is. The problem is your z limit. You went from z = 0 to z on the top surface. You need to either change your lower limit to z on the bottom surface or double your answer because of the symmetry.

 

[stratusfactio]

THANK YOU SO MUCH! I now see hwere I went wrong. I just reattempted the problem using your advice and changed the z limits from (-sqrt(4-r^2),sqrt(4-r^2)) and got the answer I was supposed to get.

Thanks again for your help and speedy response!