Elliptic Line Integral: Solving for Circulation Around an Ellipse

[terhorst]

Homework Statement

Let C be the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Evaluate $$\oint_C xdy - ydx$$.

Homework Equations

I solved this two ways. First I parameterized x and y as $$x=a \cos \theta$$ and similarly for y. I also applied Green's theorem, which yielded $$\oint_C xdy - ydx = 2 \int \int_D dA$$ where D is the area enclosed by C (ie an ellipse.) In both cases I got the answer $$2\pi a b$$.

The Attempt at a Solution

My only question is, the book I am using says the answer is $$\frac{\pi a b}{2}$$. This is an ETS book and they don't usually have typos, especially when it's the answer key to a previously administered exam. What am I missing?

 

[Dick]

The answer is pi*a*b/2. If a=b=r then it's a circle and the area is pi*r^2. So the contour is half that.

 

[terhorst]

I apologize for being so dense, but I'm still confused. A couple different books I have print the result

$$\frac{1}{2}\oint_C -ydx + xdy = \iint_{R} dA = A$$

If the area of the ellipse is $$A=\pi a b$$ then I would think that the value of the line integral is $$2A$$.

 

[Dick]

Sorry, yes, I think you are right. Don't know what I was thinking...