1. ## Volume integral

Let V be the region bounded by the planes x=0, y=0 and z=2, and the surface $\displaystyle z=x^2+y^2$ which is a paraboloid, lying in the quadrant $\displaystyle x \geq 0, y \geq 0$. Integrate $\displaystyle \iiint yz dxdydz$, choosing the appropriate limits, first in the order dx dy dz and then in the order dz dy dx, showing that the answer comes out the same both times. Then change to cylindrical polar coordinates and do the same".

Basically I have tried this and the outcome is wrong. I think I am choosing the limits wrong but I can't see why. For the first I did: z between 0 and $\displaystyle x^2+y^2$, y between 0 and $\displaystyle \sqrt{2-x^2}$ and then x between 0 and $\displaystyle \sqrt{2}$. For the second, I did: x between 0 and $\displaystyle \sqrt{z-y^2}$, y between 0 and $\displaystyle \sqrt{z}$ and z between 0 and 2. Forget the cylindrical polars for the time being, I would like to first get to grips with the normal coordinates versions. Please can someone tell me whether any of what I did is right, and if so, which parts are wrong? What should the limits of integration be?

2. ## Re: Volume integral

The paraoloid $\displaystyle z= x^2+ y^2$ crosses the plane z= 2 where $\displaystyle x^2+ y^2= 2$. That projects to the xy-plane as a circle with center (0, 0) and radius $\displaystyle \sqrt{2}$. Since we are only working in the first quadrant, it is a quarter circle and we have $\displaystyle y= \sqrt{2- x^2}$. That is, x goes from 0 to $\displaystyle \sqrt{2}$ and, for each x, y goes from 0 to $\displaystyle \sqrt{2- x^2}$. Of course, for each x and y, z goes from $\displaystyle x^2+ y^2$ to 2.

The integral is $\displaystyle \int_{x= 0}^{\sqrt{2}}\int_{y=0}^{\sqrt{2- x^2}}\int_{z= x^2+ y^2}^2 yz dzdydx$.

3. ## Re: Volume integral

So, I had the limits for y and x correct, but the initial y limits wrong. I had the answer as $\displaystyle \frac{2\sqrt{2}}{5}$. However, I later arrived at $\displaystyle \frac{16\sqrt{2}}{21}$. I troubled myself when I came out with different answers for each set of limits.