• Support PF! Buy your school textbooks, materials and every day products Here!

Using Green's Theorem

  • Thread starter khemist
  • Start date
  • #1
254
0

Homework Statement


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

0 [tex]\leq \phi \leq 2\pi[/tex]

Homework Equations



I used Green's Theorem:

[tex] \oint_C xdy - ydx[/tex]

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:

[tex] \frac {3a}{2} \int_0^{2\pi} cos^2\phi sin^2\phi d\phi[/tex]

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

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

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

The latex on my computer isnt working, but hopefully its working on everyone else's?
 

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,720
16
I get the coefficent outside the integral to be 3a^{2} but apart from that I see nothing wrong with this working.
 

Related Threads for: Using Green's Theorem

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
5
Views
1K
Replies
7
Views
533
Replies
1
Views
556
Replies
3
Views
540
Replies
4
Views
6K
Replies
9
Views
4K
Replies
4
Views
1K
Top