Thread: Evaluate a triple integral, given a bounded region G

This one really makes me scratch my head; it doesn't help that my textbook omits any details relating to triple integrals, claiming "use details from double integrals". Talk about being cheated.

Let be the region in bounded by , and . Evaluate the integral:


I don't know how this one is supposed to work, so I'll need help.

First, you need to draw the picture and determine the bounds of integration. Have you done this yet?

Can you visualize the solid over which you're integrating? I would utilize a lot of symmetry in this problem. It will make your problem much simpler.

This one isn't exactly easy to visualize (at least for me), though if I had to make a wild guess at the moment, I'd guess that the bounds of integration would be as follows:



That's all I'm drawing at this second; my brain isn't exactly in "math-mode" at the moment.

EDIT: I have a few other details from asking around.

If we concentrate on the first octant, the answer will be equal to 4 times the value of the integral in the first octant (due to symmetry). (unfortunately, my textbook and notes don't talk about octants, or not in those words, at least)

In the first octant, .

is a plane, and and are paraboloids, although I'm not sure where to go from here.

I agree with your break-down. I would further break it down in the xy-plane between those values on one side of the line y = x, and those values on the other.

Your solid kind of looks like a cathedral ceiling upside-down (maybe one intersection).

Originally Posted by Runty

This one really makes me scratch my head; it doesn't help that my textbook omits any details relating to triple integrals, claiming "use details from double integrals". Talk about being cheated.

Let be the region in bounded by , and . Evaluate the integral:


I don't know how this one is supposed to work, so I'll need help.

This plot by Wolfram Alpha may help you visualize this.

Plot by Wolfram Alpha