Calculus 3 - Stuck on Green's Theorem Problem

[wisconsin325]

I am trying to calculate the following integral using Green's Theorem:

\(\displaystyle \int_C\, \left[(3x\, +\, 5y)\hat{i}\, +\, (2x\, +\,7y)\hat{j}\right]\, dr\)

C is the circular path with a center at (2,3) and a radius of 2, oriented counterclockwise.

I started off by calculating the partial derivatives of F and then subtracted the partial of the component of of i in respect to y by the component of j in respect to x. I came up with -3, and so I thought the area of the region would be the area of the region enclosed by the circular path, which would be (pi)(2^2) or 4pi.

I proceeded to multiply -3 by 4pi and obtained -12pi as a final answer. I am unsure as to whether I took a wrong turn after coming up with the -3.

Any help would be greatly appreciated, I've been struggling with this problem. Thank you!

 

[galactus]

I assume that is a typo and you meant 2x+7y?.

I would think so. Anyway, it would appear you're on the right path....get it?...right path!

That was bad. I'm sorry :wink:

 

[wisconsin325]

That was a typo! Sorry! Thanks for the quick response . I was just wondering how the center factored into the line integral in this problem, that's where I get confused.

 

[galactus]

The circle is still a circle rather centered at the origin or at (2,3).

Your circle has equation \(\displaystyle (x-2)^{2}+(y-3)^{2}=4\)

\(\displaystyle -3\int_{0}^{2\pi}\int_{0}^{2}rdrd{\theta}=-12{\pi}\)

The area of the circle can be found using Green's theorem:

\(\displaystyle \frac{1}{2}\oint_{C}-ydx+xdy\)

\(\displaystyle x=2cos(t)+2, \;\ y=2sin(t)+3\)

\(\displaystyle \frac{dx}{dt}=-2sin(t), \;\ \frac{dy}{dt}=2cos(t)\)

\(\displaystyle \frac{1}{2}\int_{0}^{2\pi}\left[-(2sin(t)+3)(-2sin(t))+(2cos(t)+2)(2cos(t))\right]dt\)

\(\displaystyle =\frac{1}{2}\int_{0}^{2\pi}\left[4cos(t)+6sin(t)+4\right]dt=4{\pi}\)

The area of the circle.

 

[wisconsin325]

Your help is greatly appreciated Galactus. Thanks you and take care!

 

[galactus]

I just edited the post and added the area Green's theorem, and you're welcome.