1. ## Triple integral problem

I need to set the limits of integration for :

The volume of the region between z = x and the surface z = x^2 and the planes y = 0 and y = 3

I am having trouble figuring these bounds!! I know that y goes from 0 to 3. Would z be bound from x to x^2 and x from square root of z to z?????

Any help would much appreciated. Frostking

2. If you have a volume bounded by $g(x,y)\leq z \leq f(x,y)$ then you can calculate the volume as $\iint_D (f(x,y)-g(x,y))dxdy$ where $D$ is the projection on the xy-plane.

We already know the limits for y, and $z=x=x^2 \Longleftrightarrow x=0,\ x=1$

3. Originally Posted by Frostking
I need to set the limits of integration for :

The volume of the region between z = x and the surface z = x^2 and the planes y = 0 and y = 3

I am having trouble figuring these bounds!! I know that y goes from 0 to 3. Would z be bound from x to x^2 and x from square root of z to z?????

Any help would much appreciated. Frostking
setting the equations for z equal we get

$x=x^2 \iff x^2-x=0 \iff x(x-1)=0$ so $x =0 \mbox{ or } x=1$

$\int_{0}^{1} \int_{0}^{3} \int_{x^2}^{x}f(x,y,z)dzdydx$

Since z only depends on x we could switch the order of the x and y integration becuase they don't depend on each other

$\int_{0}^{3} \int_{0}^{1} \int_{x^2}^{x}f(x,y,z)dzdxdy$