# Evaluate a triple integral, given a bounded region G

• Dec 3rd 2010, 03:04 PM
Runty
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 $G$ be the region in $R^3$ bounded by $z=x^2,\ z=y^2$, and $z=4$. Evaluate the integral:
$\iiint_G |x|dV$

I don't know how this one is supposed to work, so I'll need help.
• Dec 3rd 2010, 03:48 PM
dwsmith
First, you need to draw the picture and determine the bounds of integration. Have you done this yet?
• Dec 3rd 2010, 03:55 PM
Ackbeet
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.
• Dec 3rd 2010, 04:29 PM
Runty
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:
$-2\leq x\leq 2$
$-2\leq y\leq 2$

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, $|x|=x$.

$z=4$ is a plane, and $z=x^2$ and $z=y^2$ are paraboloids, although I'm not sure where to go from here.
• Dec 3rd 2010, 06:00 PM
Ackbeet
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).
• Dec 3rd 2010, 06:29 PM
SammyS
Plot for triple integral
Quote:

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 $G$ be the region in $R^3$ bounded by $z=x^2,\ z=y^2$, and $z=4$. Evaluate the integral:
$\iiint_G |x|dV$

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