Calculating Average Value of f(x,y,z) in Solid Bounded by Cylinders

[VinnyCee]

Here is the problem:

First Part (already done): Find the volume of the solid that is bounded above by the cylinder $$z = 4 - x^2$$, on the sides by the cylinder $$x^2 + y^2 = 4$$, and below by the xy-plane.

Answer: $$\int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;dz\;dy\;dx\;=\;12\pi$$

Using the integral worked out above, and assuming that $$f\left(x, y, z\right) = \sqrt{x\;y\;z}$$. Setup the integral to find the average value of the function within that solid.

Here is what I have:

$$\frac{1}{12\pi}\;\int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;\sqrt{x\;y\;z}\;dz\;dy\;dx$$

Does that look right?

 

[Theelectricchild]

Assuming your first integral is correct, which by initial inspection, I believe it is, then yes, your second solution is indeed correct.

BTW so many of your posts have been about multivar. calc! I can tell you might have a pretty big exam coming up!

:D

Wait wait nm, I meant

D:

 

[VinnyCee]

Big exam coming

Indeed, I do have a final coming up in about a week! However, I only have one more problem to check here and then I will be concentrating on my Research Paper and Presentation for the rest of this week and upcoming weekend. It is supposed to be about superconductors and their future applications in computing. I have to present on Monday.