# Problem with spherical coordinates

• December 7th 2009, 03:53 AM
aplrt
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
• December 7th 2009, 05:03 AM
HallsofIvy
Your problem is that $\rho$ does NOT go from 0 to 1. (And that is why cylindrical coordinates is a much better choice than spherical coordinates). In spherical coordinates, $\rho$ is the straight line distance from (0,0,0) to (x,y,z) which is, of course, [tex]\sqrt{x^2+ y^2+ z^2}[tex]. For any point on the parabola $z= 1- x^2-y^2$, $x^2+ y^2+ z^2= x^2+ y^2+ (1- x^2- y^2)^2$ which is, if my algebra is correct, $x^2+ y^2+ (1- 2x^2- 2y^2- 2x^2y^2+ x^4+ y^4)$ $= 1- x^2-y^2+ (x^2- y^2)^2$. In spherical coordinates, $x= \rho cos(\theta)sin(\phi)$ and $y= \rho sin(\theta)sin(\phi)$ so that becomes $\rho^2= 1- \rho^2 cos^2(\theta)sin^2(\phi)- \rho^2 sin^2(\theta)sin^2(\phi)+ \rho^4 cos(2\theta)sin^2(\phi)$.

Solve that equation (quadratic in $\rho^2$; since $\rho$ is never negative, you can discard one root) for $\rho$ and use that as the upper limit of the integral.

I said cylindrical coordinates was a better choice! You have plenty of circular symmetry but no spherical symmetry.
• December 7th 2009, 06:17 AM
aplrt
I guessed it must have something to do with r, but it actually took me a while to understand why r does not go from 0 to 1 (I was thinking of the graph as a sphere rather than a parabola...).

Thanks for the help ! :)