Integration of multiple variables

[ElijahRockers]

Homework Statement

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

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?

 

[Dick]

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

 

[ElijahRockers]

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

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

 

[Dick]

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

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.

 

[ElijahRockers]

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

I had 8x-xy-y^2.

 

[Dick]

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

I had 8x-xy-y^2.

I think it's 8y-xy-y^2. Are you changing your answer? I'm not.

 

[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

 

[Dick]

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

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.

 

[ElijahRockers]

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.

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)

 

[Dick]

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)

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.

 

[ElijahRockers]

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.

...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.

 

[Dick]

...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.

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.