# Integration of multiple variables

1. Mar 7, 2012

### ElijahRockers

1. The problem statement, all variables and given/known data

Find the volume of the given solid.
Under the plane x − 2y + z = 8 and above the region bounded by x + y = 1 and x2 + y = 1

3. The attempt at a solution

Here's how I set it up.

$\int^1_0 \int^{1-x^2}_{1-x} (8-x+2y) dydx$

When I do the math, I get 21/20. I have gone several different routes using a calculator and I keep getting that answer. The software tells me the answer is 29/20.

So am I setting it up wrong?

2. Mar 7, 2012

### Dick

For what it's worth, I get 21/20 as well.

3. Mar 7, 2012

### ElijahRockers

Well, I copied and pasted the question directly from the software this time. Stupid software.

4. Mar 7, 2012

### Dick

Well, assuming we are right, I doubt the software figured out the answer. Somebody told it what the answer should be. They are to blame.

5. Mar 7, 2012

### ElijahRockers

Hmm I found an error in my work. The anti derivative of 8-x+2y is 8x-xy+y^2.

6. Mar 7, 2012

7. Mar 7, 2012

### ElijahRockers

But if x-2y+z=8, then z=8-x+2y dz/dy = 8y -xy +y^2. I haven't gone through the problem, I did a similar one with different numbers and got it right

8. Mar 7, 2012

### Dick

You are integrating z, not differentiating it. I thought you did go through it and got 21/20?? I'm not sure where you are going with this. I think you did it right the first time.

9. Mar 7, 2012

### ElijahRockers

Oh yeah, sorry, by dz/dy i mean the anti derivative.

And yes, i got 21/20 when i used 8x-xy-y^2. I noticed my error, but by that time had already gotten the question right with a different set of values.

I posted my error here just incase you were wondering, and to see if, by some bizarre one in a million coincidence, you made the same exact error I did. (not likely, especially considering you are probably much more careful than I am)

10. Mar 7, 2012

### Dick

Wow. Apparently I'm not much more careful than you. I made a completely different error. I integrated z=8-x-2y instead of z=8-x+2y and got 21/20. Figured since I got the same as you did, no need to double check. Oooops. Yeah, the odds are pretty low for this to happen.

Last edited: Mar 7, 2012
11. Mar 7, 2012

### ElijahRockers

........Actually, the odds drop drastically when we consider that you actually didn't make a completely different error, but you made the same exact error I did, completely separately from me. Look at the post where I mentioned my error.

:surprised

12. Mar 7, 2012

### Dick

Ok, yeah. I get you, it's wasn't the 8x instead of the 8y, it was the sign on the 2y. That does cut the odds.