Setting up triple integrals in different coordinates

[anna.schweizer]

Homework Statement

Assume that f(x,y,z) is a continuous function. Let U be the region inside the cone z=√x^2+y^2 for 2≤x≤7. Set up the intregal ∫f(x,y,z)dV over U using cartesian, spherical, and cylindrical coordinates.

Homework Equations

CYLINDRICAL COORDINATES

• x=rcosθ
• y=rsinθ
• z=z

SPHERICAL COORDINATES

• ρ^2 =x^2 + y^2 + z^2
• x = ρsin(phi)cos(θ)
• y=ρsin(phi)sin(θ)
• z=ρcos(phi)

The Attempt at a Solution

Do Ineed to have a function for the inner two integrals?

For my limits on integration for Cartesian coordinates are
z = x^2 + y^2 -4 and z = x^2 + y^2 -25
I DON'T KNOW FOR X
and y = 2 and y = 5

For my limits on the spherical coordinates are
θ=2π, θ=0; and ρ=2, ρ=5

I DON'T KNOW IF WHAT I'M DOING IS RIGHT OR NOT. I'M LOST :(

Please and thank you for your help!

 

[mfb]

Do Ineed to have a function for the inner two integrals?

Your function is just f, you can use it as f(r, θ, z) and similar for spherical coordinates, that is fine.
Your task is to find the integral borders and to convert dV.

I DON'T KNOW FOR X

That is given in the problem statement, and please don't write in caps.

Your attempt at a solution uses equations different from the problem statement.