# Volume by nested integrals

• Feb 28th 2009, 12:22 PM
Frostking
Volume by nested integrals
I need to determine the volume of the region between the plane z= x and the surface z = x^2 and the planes y = 0 and y = 3. I think the outer integral would be for dy and 0 and 3 the bounds but I am not sure about much else, especially what do I use for my function??? x^2??
• Feb 28th 2009, 01:51 PM
Jester
Quote:

Originally Posted by Frostking
I need to determine the volume of the region between the plane z= x and the surface z = x^2 and the planes y = 0 and y = 3. I think the outer integral would be for dy and 0 and 3 the bounds but I am not sure about much else, especially what do I use for my function??? x^2??

$\displaystyle z = x,\;\;z=x^2$ are the two respective heights of the surfaces (they intersect at $\displaystyle x=0,\, x = 1$). If you draw them in the x-z plane you will see this. You are correct on your outer integral, the full integral is

$\displaystyle V = \int_0^3 \int _0^1 x - x^2 dx\,dy$