Math Help - Integration first converting to spherical polars

1. 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?

2. 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 $x = r \cos(\varphi) \sin (\theta)$, and $y = r \sin(\varphi) \sin(\theta)$. You should use this definition to calculate $(x^2 + y^2)^{\frac{1}{2}}$, and you should get $(x^2 + y^2)^{\frac{1}{2}} = r \sin(\theta)$

Giving you an integrand of $\frac{r \cos(\theta)}{r \sin(\theta} r^2 \sin(\theta) = r^2 \cos(\theta)$

3. That's one reason why it is more common to use " $\rho$" 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 $x= \rho cos(\phi) sin(\theta)$ and $y= \rho sin(\phi) sin(\theta)$. Then $x^2+ y^2= \rho^2 cos^2(\phi)sin^2(\theta)+ \rho^2 sin^2(\phi)sin^2(\theta)$ $= \rho^2 sin^2(\theta)(cos^2(\phi)+ sin^2(\phi))= \rho^2 sin^2(\phi)$ so that $(x^2+ y^2)^{1/2}= \rho sin(\phi)$