# Thread: Greens Theorem

1. ## Greens Theorem

Find $[integral](x^2)*(y^2)dx + 4*x*y^3dy$ where C is the boundary of the triangle with the corners (0,0), (1,3), and (0,3). Mainly how do I find the limits of integration?

2. Originally Posted by Sgraveyard
Find $[integral](x^2)*(y^2)dx + 4*x*y^3dy$ where C is the boundary of the triangle with the corners (0,0), (1,3), and (0,3). Mainly how do I find the limits of integration?
Did you draw the picture?

3. Why do you need Green's theorem?

4. Originally Posted by maddas
Why do you need Green's theorem?
The problem is a path integral around a closed path. It would not be difficult to integrate but, since the path is not smooth would require three different integrals. Green's theorem allows that to be done as a single integral over the region enclosed by the path.

Sgraveyard, as dwsmith suggested, first draw the picture. It is, of course, a triangle having (0, 0), (1, 3), and (0, 3) as vertices.

The limits of integration depend on the order of integration. If you decide to integrate with respect to y first, then x, you need to look at the smallest and largest possible values for x. They are, of course, 0 and 1. Now, draw a vertical line representing "x= constant". For each x, that line crosses the triangle at y= 3 above and on the line from (0, 0) to (1, 3) below. Write the equation for that line and solve for y. That will be the lower bound on the y-integral:
$\int_{x= 0}^1 \int_{y= ax+b}^3 dydx$
(You need to put in the "a" and "b".)

Or you could decide to integrate with respect to x first and then y, look at the smallest and largest values of y- they are clearly 0 and 3. Draw a horizontal line, representing y= constant. That will cross the triangle at x= 0 and on the line from (0,0) to (1, 3) on the right. Write the equation for that line and solve for x. That will be the upper bound on the x-integral:
$\int_{y= 0}^3\int_{x= 0}^{my+ c} dxdy$