Area under a curve described by parametric equations.?

Question

I want to find the area under a curve described by parametric equations x = f(t) and y = g(t). I can show that the area is the integral of g(t)f'(t)dt (from t1 to t2), but does this work for a curve that fails the vertical line test? That is, can I use the formula to find the area of a circle desribed parametrically? (x=cost, y=sint) or some other function where x'(t) is not always positive? It doesn't always seem to work for me, but maybe I'm missing something.

Answer 1

The direct answer to your question is "yes, you can".
The sign of x'(t) really doesn't matter.
Even for the area inside a closed circle, you can get the right answer with the formula:

int(sin(t)cos'(t), t=0..2Pi) = int(-sin^2(t), t=0..2Pi)
= - Pi.

There's a negative sign because it's integrated counter clockwise.

Answer 2

I think you will find this link very helpful. Just click area under a parametric.

http://www.math.odu.edu/cbii/calcanim/

Regards,

Mysstere