Let a, b0, and consider the ellipse parametrized by: x=a cos t and y= b sin t for 0

Question

Let a, b>0, and consider the ellipse parametrized by:

x=a cos t and y= b sin t for 0<=t<=2pi

a. Find an integral that represents the circumference C_ab of the ellipse. 

I keep making the same mistake, when I plug in f'(x) and g'(x).

I got f'(x) = (a(-sint) + cost) and g'(x) = (bcost + sint)

when I plug this into the Length integral, I am off by a little and I can't figure out where the mistake was made. Please help!

Thank you!

anonymous · Answer

@hanifah I guess its asking for the perimeter of the ellipse, am I right?

anonymous · Answer

$$L=\int\limits_{0}^{2\pi}rdt$$
$$r=\sqrt{x^2+y^2}=\sqrt{a^2\cos^2(t)+b^2\sin^2(t)}\rightarrow L=\int\limits_{0}^{2\pi}\sqrt{a^2\cos^2(t)+b^2\sin^2(t)}dt$$

anonymous · Answer

Woah, that's it? So I don't even need to find the derivative of x or y?
Thanks so much @CarlosGP

anonymous · Answer

Circumference doesn't equal perimeter, but that's the answer in the back of my book

anonymous · Answer

you are welcome
Circumference is a special case of perimeter in that the perimeter is typically around a polygon while circumference is around a closed curve.

anonymous · Answer

I see, that makes sense!