Question

If a circle C with radius 1 rolls along the outside of the circle x^2+y^2=16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x=5cos(t)-cos(5t), y=5sin(t)-sin(5t). Find the area it encloses.

Answer 1

Have you learned about polar coordinates?

Answer 2

Yes

Answer 3

We are doing line integrals and greens theorem now though. Is there some way I could solve it using Green's Theorem?

Answer 4

Do you remember what greens theorem states?

Answer 5

\[A=\int\limits_{C}^{}x \ dy=-\int\limits_{C}^{}y \ dx= \frac{ 1 }{ 2 }\int\limits_{C}^{}x \ dy \ - y\ dx\]

Answer 6

Ah, that's easy then.

Answer 7

\[
\oint_C xdy = \int_a^bx\frac{dy}{dt}dt
\]Where \(a,b\) are the begining and ending of the parametrization

Answer 8

Thanks!

Answer 9

I remember what to do from here

Answer 10

Ok actually, I know how to do this just plug in and such so that I have:
But that will take forever to solve. Is there an easier way to solve this?

Answer 11

I don't think your xdy/dt is correct

Answer 12

I just checked and mathematica says that the dy/dt part I had was right?

Answer 13

\[
(5\cos(t)-\cos(5t))\cdot (5\cos(t)-5\cos(5t))
\]

Answer 14

Yep that's what I have

Answer 15

What's so hard about that integral?

Answer 16

You can do either one, but the first one seems easiest

Answer 17

I got it thanks!