Calculate Mass of a Sphere w/ Density f(ρ,ϕ,θ) = 2e^-p^3

[mharten1]

Homework Statement

Find the mass of the following object with the given density function:

The ball of radius 8 centered at the origin with a density f(ρ,ϕ,θ) = 2e^-p^3

Homework Equations

The Attempt at a Solution

I know that I need to integrate the density function, but I'm not sure what limits to use. My book tells me that a sphere with radius a centered at the origin has the description: {(ρ,ϕ,θ): p=a}, a>0. However, I'm not exactly getting how this helps me find the limits.

 

[LCKurtz]

Homework Statement

Find the mass of the following object with the given density function:

The ball of radius 8 centered at the origin with a density f(ρ,ϕ,θ) = 2e^-p^3

Homework Equations

The Attempt at a Solution

I know that I need to integrate the density function, but I'm not sure what limits to use. My book tells me that a sphere with radius a centered at the origin has the description: {(ρ,ϕ,θ): p=a}, a>0. However, I'm not exactly getting how this helps me find the limits.

Do you understand spherical coordinates? Can you describe the ranges the three variables $\rho,\ \phi,\ \theta$ must traverse to sweep out a sphere of radius $a$ centered at the origin?

 

[mharten1]

Do you understand spherical coordinates? Can you describe the ranges the three variables $\rho,\ \phi,\ \theta$ must traverse to sweep out a sphere of radius $a$ centered at the origin?

Would the limits be:

for ρ 0 to 8

for θ 0 to 2pi

and for ϕ 0 to pi?

If not, I need to review spherical coordinates.

 

[LCKurtz]

Would the limits be:

for ρ 0 to 8

for θ 0 to 2pi

and for ϕ 0 to pi?

That's right, and you just answered your own question about what limits to use.

 

[mharten1]

That's right, and you just answered your own question about what limits to use.

Thank you. Sometimes when you have a lot on your mind it's easy to forget simple things. :P