# Homework Help: Converting to Spherical Coordinates then integrating? Am I doing this right?

1. May 29, 2012

### emzee1

Converting to Spherical Coordinates....then integrating? Am I doing this right?

1. The problem statement, all variables and given/known data
Consider the integral ∫∫∫(x2z + y2z + z3) dz dy dx, where the left-most integral is from -2 to 2, the second -√(4-x2) to √(4-x2) and the right-most integral is from 2-√(4-x2-y2) to 2+√(4-x2-y2)

2. Relevant equations
x=ρsin(phi)cos(θ)
y=ρsin(phi)sin(θ)
z=ρcos(phi)
x2+y2+z22

3. The attempt at a solution
So I need to convert to spherical coordinates. The first 2 integrals describe a region on the xy-plane that's a circle, centered at the origin, with a radius of 2. The distance from the origin I thought was from 0 to 4, as described the right-most integral from above. Then using the equations from above, I converted the integrand:

∫∫∫ρcos(phi)ρ2sin(phi) dρ d(phi) dθ

Left-most integral: 0 to 2*pi
Middle: 0 to pi/2
Right: 0 to 4

Final answer = 2144.67 (which does not feel right)

I'm fairly certain I am correct for the most part, except the right most integral in the spherical-converted integral. Anyone care to check if I did everything correctly?

2. May 29, 2012

### SammyS

Staff Emeritus
Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

Hello emzee1. Welcome to PF !

The volume element in spherical coordinates is $dV=\rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta\ .$

The integrand: $x^2z+y^2z+z^3=(x^2+y^2+z^2)z \ \to\ \rho^2\left(\rho\cos(\phi)\right)=\rho^3\cos(\phi)\ .$

3. May 29, 2012

### emzee1

Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

Thanks for that SammyS, I guess that was a mental mistake on my part. What about the limits for the integral for the spherical-integral? I don't think I did those correctly...

4. May 29, 2012

### SammyS

Staff Emeritus
Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

Look at the limits of integration for z.

They describe a sphere of radius 2, centered at (x, y, z) = (0, 0, 2) .

Write that equation in spherical coordinates.

5. May 29, 2012

### emzee1

Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

So I converted the equation of the sphere:

x2+y2+(z-2)2 = 4

to:

ρ2-4ρcos(phi) = 0

solving for ρ:

ρ= 4cos(phi)

so the integral limits, in terms of dρ: 0 to 4cos(phi) ?
Then the limits of d(phi): 0 to pi/2
And the limits of dθ: 0 to 2*pi?

Does this sound correct?

6. May 29, 2012

### SammyS

Staff Emeritus
Re: Converting to Spherical Coordinates....then integrating? Am I doing this right?

That looks better --- correct.

I'm notorious for overlooking details! LOL !