Multiple Integrals: Find Volume Bounded by Cylinders and Planes

[bodensee9]

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

 

[HallsofIvy]

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, [itex]y= \sqrt{1- x^2}[/itex]. Finally, for each x and y, z runs from 0 up to the plane z= y. The volume is given by
[tex]\int_{x=0}^1\int_{y=0}^{\sqrt{1-x^2}}\int_{z=0}^y dzdydx= \int_{x=0}^1\int_{y=0}^{\sqrt{1-x^2}}y dydx[/itex]
No, that is NOT [itex]\sqrt{1- x^2}dydx[/itex]! 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!

 

[bodensee9]

Oh I see now! Thanks!