# Thread: Green's theorem and line integral problem

1. ## Green's theorem and line integral problem

Use Green's theorem to evaluate

$\int_{C} 3xydx+2x^{2}dy$

Where C is the positively oriented boundary bounded by the parabola y=x^2 and y=-x

I think I've got it set up right, but the end result is 0. Is that correct?

2. ## Re: Green's theorem and line integral problem

You think you've set it up right? Good, what did you do to set it up?

3. ## Re: Green's theorem and line integral problem

Originally Posted by Prove It
You think you've set it up right? Good, what did you do to set it up?
I took the partial derivatives of 3xy (with respect to y) and 2x^2 (with respect to x), so the integral to solve is:

$\int_0^4 \int_{x^{2}+2x}^{-x} x {d}y {d}x$

And I ultimately get zero.

4. ## Re: Green's theorem and line integral problem

I agree with your integrand but not your boundaries. Have you tried sketching the region?

5. ## Re: Green's theorem and line integral problem

Originally Posted by Prove It
I agree with your integrand but not your boundaries. Have you tried sketching the region?
That's because I wrote the boundaries wrong.

The boundaries are:

y= -x
$y=x^{2}+2x$

6. ## Re: Green's theorem and line integral problem

Your boundaries are still wrong. Where do these graphs intersect?

7. ## Re: Green's theorem and line integral problem

Man it way too late for my dumb stuff.

Here is the fixed integral:

$\int_{-4}^0 \int_{x^{2}+2x}^{-x} x {d}y {d}x$

Now THAT should be the correct region and should add up to zero.......maybe?

8. ## Re: Green's theorem and line integral problem

Surely you can see that the graphs of \displaystyle \begin{align*} y = x^2 + 2x \end{align*} and \displaystyle \begin{align*} y = -x \end{align*} do NOT intersect at x = -4...

9. ## Re: Green's theorem and line integral problem

Originally Posted by Prove It
Surely you can see that the graphs of \displaystyle \begin{align*} y = x^2 + 2x \end{align*} and \displaystyle \begin{align*} y = -x \end{align*} do NOT intersect at x = -4...
Thank you, you're right. They intersect at x=-3.

So if I just change the interval everything else is correct, yes?

10. ## Re: Green's theorem and line integral problem

Well, except the result, I need to reevaluate that now

11. ## Re: Green's theorem and line integral problem

Yes if you change the -4 to -3 your boundaries are now correct, so now you can correctly evaluate the line integral...

12. ## Re: Green's theorem and line integral problem

Originally Posted by Prove It
Yes if you change the -4 to -3 your boundaries are now correct, so now you can correctly evaluate the line integral...
Thank you ProveIt, I am going to go to bed now but will clean it up tomorrow and post the answer.

I appreciate your help and teaching style......much better than just someone telling it's right/wrong/giving me the answer!

13. ## Re: Green's theorem and line integral problem

You're welcome I tend to be the most help to people who help themselves, so when someone has posted everything they've done it's very easy to point out what needs to be worked on

14. ## Re: Green's theorem and line integral problem

You said in your original post that "C is the positively oriented boundary bounded by the parabola y=x^2 and y=-x".

So where did x^2+ 2x come from?

15. ## Re: Green's theorem and line integral problem

The OP explained in Post 5 that the original post had a typo and posted the correct bounding functions.

Page 1 of 2 12 Last