Thread: Integration first converting to spherical polars

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)

So,

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?

Originally Posted by StarWrecker

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)

So,

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?

Why did you choose to ignore z?

In spherical coordinates , and . You should use this definition to calculate , and you should get

Giving you an integrand of

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