That is a large value your professor got. I am not so sure he/she is correct.
I need some help with a triple integral problem that I keep getting wrong in the last step somehow.
the problem is:
Evaluate on E xyz dV where E lies between the spheres =2 and =4 and above the cone = /3.
There work I've done so far is set up the integral integrated it and got:
1/4 sin^4 from 0 to * 1/2sin^2 from 0 to 2 * 1/6 ^6 from 2 to 4.
I get that the middle term should be zero, but for some reason my prof gives the answer to be
Consider symmetry. The region of integration is rotationally symmetric about the z axis, and so if we rotate the function being integrated about the z axis, the integral should be unchanged. Now, let us rotate the integrand by 90 degrees about the z axis: we transform x and y to x' and y' via x'=-y and y'=x. Then the integrand is f(x',y',z)=xyz=(y')(-x')z=-x'y'z.
Thus, we have . But, if we rename the variables x' and y' as x and y, (and as E'=E), we have
which means .
Thus, due to the symmetry, the integral is zero.
[We can also look at this as the fact that xyz's dependence on the azimuthal angle is of the form , and the region has no dependence on the azimuthal angle (the rotational symmetry), so the integral over will be 0]