Calculating Triple Integrals in Mathematica

[november1992]

Homework Statement

Evaluate ∫∫∫\sqrt{x^{2} + y^{2}} dA where R is the region bounded by the paraboloid y=x^2+z^2 and the plane y=4

Homework Equations

I believe this is a problem where cylindrical coordinates would be useful

0 ≤ z ≤ \sqrt{4-x^2}
0 ≤ r ≤ 2 ( I think this is wrong).
0 ≤ θ ≤ 2\pi

The Attempt at a Solution

Integrate[ r^2, {\[Theta], 0, 2*Pi}, {r, 0, 2}, {z, 0, Sqrt[4 - r*Cos[\[Theta]]^2]}]

There isn't an output when I enter this line. I checked for the syntax on the wolfram reference page so I think the problem is with the limits of integration.

 

[SteamKing]

What is the source of your integral? What is it supposed to represent? I say this because you have a triple integral with dA rather than dV.

 

[november1992]

This problem was given to me by my instructor.

 

[SteamKing]

Yeah, I figured it wasn't revealed to you in a dream. But what is the ultimate source of this integral? Is it a problem out of a book? Have you solved faster than light travel? Is a new mathematics about to be revealed to us?

You realize, I hope, that there seems to be something missing: You have a triple integral of a function with respect to what appears to be a differential element of area dA. Three integrations, two variables with respect to which the integration could be carried out. It's not clear how this integral is supposed to be calculated.