Describing D is Green's Theorem

[TranscendArcu]

Describing "D" is Green's Theorem

Homework Statement

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.

The Attempt at a Solution

I have Q_x = 3 and P_y = 0. Therefore Q_x - P_y = 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.

 

[I like Serena]

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?

 

[TranscendArcu]

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

 

[I like Serena]

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

 

[TranscendArcu]

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?

 

[I like Serena]

Yup.