multivariable calculus find the volume.?

Question

Consider the region R bounded on top by z=1-x and on the side by y=1-x^2 which lies in the first octant.

a- sketch the region R. (I'm working on it now)
b- Find the volume of the region R. ( I know i have to triple integrate but i don't know what the interval limits or integrand should be. I need to know how to set up the integration.)

thank you for your time.

Answer 1

Basically, this solid lies above the region in the xy-plane in the first quadrant and bounded by the axes and the function y=1-x². The height of this solid at any point will be 1-x. The function 1=1-x² hits the x-axis at 1, so the integration will be from x=0 to x=1. At any given x-coordinate, the integration will be from 0 to 1-x². So we have:

[0, 1]∫[0, 1-x²]∫1 - x dy dx

This integral may now be performed very simply. Note that since 1 - x is constant with respect to y, it can be pulled out of the inner integral, so we have:

[0, 1]∫(1-x)[0, 1-x²]∫1 dy dx
[0, 1]∫(1-x)(1-x²) dx
[0, 1]∫x³ - x² - x + 1 dx
x⁴/4 - x³/3 - x²/2 + x|[0, 1]
1/4 - 1/3 - 1/2 + 1
3/12 - 4/12 - 6/12 + 12/12
5/12

And we are done.

Edit: I've created a graph of the region from several different angles, which you can see here: http://img204.imageshack.us/img204/5899/severalviewsxt6.png . This should help you with your sketches.