# Describing D is Green's Theorem

1. Dec 4, 2011

### TranscendArcu

Describing "D" is Green's Theorem

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

Let F(x, y) = (tan−1(x))i+3xj. Find $$\int_C F • dr$$where C is the boundary of the rectangle with vertices (0, 1), (1, 0), (3, 2), and (2, 3), traversed counterclockwise.

3. The attempt at a solution

I have Qx = 3 and Py = 0. Therefore Qx - Py = 3 - 0 = 3. Now, what I'm having the most trouble with is just describing this rectangle in terms of x and y.

I think the boundary of my rectangle is described by line segments that follow the equations y=x+1, y=-x+5, y=x-1, and y=-x+1. I rewrite these as x=y-1, x=-y+5, x=y+1, and x=-y+1. I think I have to split up the region somehow, so my integrals are:

$$\int_0 ^2 \int_{-y+1} ^{y+1} 3 dxdy + \int_1 ^3 \int_{y-1} ^{-y+5} 3 dxdy$$

At this point, I think I should ask if I'm doing this correctly.

2. Dec 4, 2011

### I like Serena

Re: Describing "D" is Green's Theorem

Hi TranscendArcu!

Your split up regions do not appear to match your rectangle, which you can see if you would plot the outermost points of your boundaries.
2 regions would also not be enough.

But to make it a bit easier, what is in general the surface integral of a constant function, say 1?

3. Dec 4, 2011

### TranscendArcu

Re: Describing "D" is Green's Theorem

The surface integral of a constant function is the surface area of the surface (multiplied by the constant, in this case one), right?

4. Dec 4, 2011

### I like Serena

Re: Describing "D" is Green's Theorem

Yes, but only if the constant function is 1.
So....

5. Dec 4, 2011

### TranscendArcu

Re: Describing "D" is Green's Theorem

So I might just write 3*Area(D), right?

So if I find vectors that describe the edges of D, and set them in R3, and take the magnitude of their cross product, I should find the area of D. I have, <-1,1,0> and <2,2,0>. Cross product gives <0,0,-4>

|<0,0,-4>| = 4.

3*4 = 12?

6. Dec 4, 2011

### I like Serena

Re: Describing "D" is Green's Theorem

Yup.