Evaluate the following triple integral

[caesius]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

What I'm having trouble with is setting up the limits of integration.

I already have
0 < z < x
x^3 < y < 8

but what about x?

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?

 

[HallsofIvy]

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

 

[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. :)