Problem with spherical coordinates
First off, sorry I had to write all the equations in "text", but I hope it's not too impossible to read them.
I want the volume above the xy-plane, under the paraboloid z = 1 - x^2 - y^2 and in the wedge cut out by -x\< y \< sqrt(3) x . With cylindrical coordinates, I get the correct answer (=7pi/48), but with spherical coordinates I do not. Obviously I'm making some elementary error in my logic or calculations - could someone please check it out ?!
My spherical coordinates:
x = rsin(phi)cos(theta)
y = rsin(phi)sin(theta)
z = rcos(phi)
dxdydz = r^2 sin(phi) dr dphi dtheta
My tripple integral in spherical coordinates:
int(-pi/4 -> pi/3) dtheta
int(0 -> pi/2) sin(phi) dphi
int(0 -> 1) r^2 dr
According to this, I ought to get 7pi/12 * 1 * 1/3 = 7pi/36