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

[emzee1]

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

Homework Statement

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)

Homework Equations

x=ρsin(phi)cos(θ)
y=ρsin(phi)sin(θ)
z=ρcos(phi)
x²+y²+z²=ρ²

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?

 

[SammyS]

Homework Statement

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)

Homework Equations

x=ρsin(phi)cos(θ)
y=ρsin(phi)sin(θ)
z=ρcos(phi)
x²+y²+z²=ρ²

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?

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

 

[emzee1]

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...

 

[SammyS]

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...

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.

 

[emzee1]

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?

 

[SammyS]

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?

That looks better --- correct.

I'm notorious for overlooking details! LOL !