# Multiple Integrals

## Homework Statement

Hello, I was wondering if someone could help me with the following. Supposed I am asked to find the volume bounded by the cylinders x^2+y^2=1 and the planes y = z, x = 0, z = 0 in the first octant.

## Homework Equations

So this is what I tried to do. The boundaries should be: x is between 0 and 1 and y is between the squareroot of (1-x^2) and 0, or you can have y is between 0 and 1 and x is between the squareroot of (1-y^2) and 0. So wouldn't the double integral be the integral of

the squareroot of 1-x^2dydx, where you first evaluate it from 0 to the squareroot of (1-x^2), and then you evaluate it again from 0 to 1? Thanks!!!

## The Attempt at a Solution

Since you are in the first octant, yes, x runs between 0 and 1. For each x, then y runs from 0 up to the circle, $y= \sqrt{1- x^2}$. Finally, for each x and y, z runs from 0 up to the plane z= y. The volume is given by
No, that is NOT $\sqrt{1- x^2}dydx$! You don't get the square root until after integrating with respect to y- and then, since the integral of ydy will involve y2, you don't really have a square root to integrate with respect to x!