(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the volume between the plane x+y+z = 1 and the xy-plane, for x+y[itex]\leq[/itex]2, x[itex]\geq[/itex]0, y[itex]\geq[/itex]0.

3. The attempt at a solution

First, the plane is above the xy-plane for y < 1-x and below the xy-plane for y > 1-x, so we'll need two integrals. This is how I set them up.

[itex]\int[/itex][itex]^{1}_{0}[/itex][itex]\int[/itex][itex]^{1-x}_{0}[/itex][itex]\int[/itex][itex]^{1-x-y}_{0}[/itex]1dzdydx - [itex]\int[/itex][itex]^{2}_{0}[/itex][itex]\int[/itex][itex]^{2-x}_{1-x}[/itex][itex]\int[/itex][itex]^{1-x-y}_{0}[/itex]1dzdydx

This gives me an answer of 7/6, but my book has 1 as the answer. I'm assuming my setup is wrong, since I checked the evaluation of the integrals in wolfram alpha. But I can't see what I did wrong. The innermost integrals give z-value of the given plane. The middle integrals sum those stacks from 0 to 1-x above the xy-plane and from 1-x to 2-x below the xy-plane. And then the outermost integrals sum those slices from the y axis to the boundary at 0=1-x and 0=2-x respectively.

Help would be much appreciated.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Volume with triple integral

**Physics Forums | Science Articles, Homework Help, Discussion**