Volume between plane x+y+z=1 and xy-plane

[Opus_723]

Homework Statement

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

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.

$\int$$^{1}_{0}$$\int$$^{1-x}_{0}$$\int$$^{1-x-y}_{0}$1dzdydx - $\int$$^{2}_{0}$$\int$$^{2-x}_{1-x}$$\int$$^{1-x-y}_{0}$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.

 

[The_Chromer]

Weird. I tried evaluating your integrals, and I got (1/6)-(-3)=(19/6).

See what you get by evaluating the second integral again. I don't know if I made a mistake or not.

 

[The_Chromer]

Try drawing a picture of the volume that you're trying to find. When I took multi-V, I noticed that helped me visualize the question better, which is essential to setting up your integral equation.

 

[LCKurtz]

Homework Statement

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

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.

$\int$$^{1}_{0}$$\int$$^{1-x}_{0}$$\int$$^{1-x-y}_{0}$1dzdydx - $\int$$^{2}_{0}$$\int$$^{2-x}_{1-x}$$\int$$^{1-x-y}_{0}$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.

The dydx portion of your second integral needs to be broken into two sections. y goes from 1-x to 2-x only when x is between 0 and 1. When x is between 1 and 2 y goes from 0 to 2-x. Draw a picture of y = 1-x and y = 2-x in the xy plane and you will see it.

 

[The_Chromer]

Nice going.

 

[Opus_723]

The dydx portion of your second integral needs to be broken into two sections. y goes from 1-x to 2-x only when x is between 0 and 1. When x is between 1 and 2 y goes from 0 to 2-x. Draw a picture of y = 1-x and y = 2-x in the xy plane and you will see it.

AHA! Thank you! I actually had already drawn the picture, but I've been staring at it forever without noticing that! Now I feel dumb. I'm kind of in a studying binge and I think I'm starting to burn out a bit. Thank you though.