Solving for Area using Green's Theorem with Astroid Equations | Homework Help

[khemist]

Homework Statement

Use a line integral to find the area of the region enclosed by astroid
x = acos³$$\phi$$
y = asin³$$\phi$$

0 $$\leq \phi \leq 2\pi$$

Homework Equations

I used Green's Theorem:

$$\oint_C xdy - ydx$$

The Attempt at a Solution

I solved for dx and dy from my parametric equations. I then plugged in x, y, dx, and dy into the integral to solve for the area.

After simplifying, I came out with:

$$\frac {3a}{2} \int_0^{2\pi} cos^2\phi sin^2\phi d\phi$$

Now in order to solve this, I used a half angle formula, $$cos\phi sin\phi = (\frac{1}{2}sin2\phi)^2 = \frac {1}{4}sin^2 2\phi$$

Which then I used a different angle formula to get:$$\frac{1}{8}(1-cos4\phi)$$

Am I on the right track? I would then integrate to solve...

The latex on my computer isn't working, but hopefully its working on everyone else's?

 

[hunt_mat]

I get the coefficient outside the integral to be 3a^{2} but apart from that I see nothing wrong with this working.