(adsbygoogle = window.adsbygoogle || []).push({}); Converting to Spherical Coordinates....then integrating? Am I doing this right?

1. The problem statement, all variables and given/known data

Consider the integral ∫∫∫(x^{2}z + y^{2}z + z^{3}) dz dy dx, where the left-most integral is from -2 to 2, the second -√(4-x^{2}) to √(4-x^{2}) and the right-most integral is from 2-√(4-x^{2}-y^{2}) to 2+√(4-x^{2}-y^{2})

2. Relevant equations

x=ρsin(phi)cos(θ)

y=ρsin(phi)sin(θ)

z=ρcos(phi)

x^{2}+y^{2}+z^{2}=ρ^{2}

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)ρ^{2}sin(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?

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

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