# Evaluate the following triple integral

1. May 19, 2009

### caesius

1. The problem statement, all variables and given/known data
Evaluate the following triple integral

$$I = \int\int\int_{R}x dv$$

in Cartesian coordinates where R is the finite region bounded by the surfaces z=0, y=x^3, y=8, z=x. Sketch the region R. Here dV is the element of volume.
2. Relevant equations

3. The attempt at a solution
What I'm having trouble with is setting up the limits of integration.

0 < z < x
x^3 < y < 8

And how do I know that the y and z limits are that way around and not x < z < 0 and 8 < y < x^3 instead?

2. May 20, 2009

### HallsofIvy

Staff Emeritus
Those inequalities do NOT describe a bounded region. Aren't there additonal restrictions?

3. May 20, 2009

### squidsoft

Hello Caesius.

May I make a suggestion I think would be helpful to you?

Suppose all you had to do was to plot the surfaces. Never mind (for now) the integration. Could you do that, nicely? The surface z=0 is just the x-y plane right. The surface y=x^3 is a paraboloid sheet, and z=x is a diagonal flat sheet. Suppose that was the only assignment, draw these three surfaces together, transparently so you could see where they intersect, and do it nicely. Then study them, closely, rotate the figure around interactively (you can do that in Mathematica), note the intersections, then go through the algebra proving your observations, then come back and answer your question. :)