I'm trying to work through a question, but I cannot get my result to look anything like the model answers.
The question is to integrate
z/((x^2+y^2)^(1/2)) over the volume of a region bounded by the surfaces z=0 and z=(9-x^2-y^2)^(1/2).
I know I need to use spherical polar co-ordinates to do this, but I can't seem to get the substitutions right.
The model answer says that it becomes r^2*cos(theta) integrated over the obvious region, but my attempt to get it is as such:
z = r*cos(theta)
r = (x^2+y^2)^(1/2) (in 2 dimensions, ignoring z)
the integral is
integral( (r*cos(theta) / r) * r^2 * sin(theta) drd(theta)d(phi) )
Of course this gives me r^2*cos(theta)*sin(theta).
What am I missing?
That's one reason why it is more common to use " " as the radial variable in spherical coordinates (and they are NOT normally referred to as "spherical polar coordinates), to avoid confusing 3d and 2d coordinates.
In spherical coordinates and . Then so that