How to Calculate Area Enclosed by a Complex Polar Curve Using Green's Theorem?

[Niskamies]

The problem is the following:

Use a line integral to find the plane area enclosed by the curve r = a*(cos t)^3 + b*(sin t)^3, 0 < t <2*pi.

I don't really have a clue how to solve that. The chapter we are right now in is about Green's theorem in the Plane, and all the example problems use x and y instead of t.

I would be very glad i f somebody would tell how to solve this problem.

 

[chaoseverlasting]

I would integrate the curve 'r' wrt t, taking the limit to be 0 to 2pi.

 

[lalbatros]

This is easily done (I think) using polar coordinates.
If r(t) is the distance from the origin to the curve,
and if t is the polar angle around the origin,
then the surface is given by

Integral[r(t)²/2 dt,{t,tmin,tmax)]

You have to be careful to decide the limits of integration,
and therefore you need to make a correct graphic of this function to understand the shape you need to evaluate.
There may be some calucations to perform ...

Michel

Note:
On the lhs, you wrote r in bold.
Be careful, r is the distance in polar coordinates, it is not a vector and should not be written in bold.