Compute area using divergence and flux?

[nebbie]

Compute area using divergence and flux??

Consider the curve given by g(t) =acos^3(t),asin^3(t), where t is [0; 2pi] and a > 0 is a constant.

(a) Find the unit tangent and outward normal vectors.
(b) Compute the area enclosed by this curve.

I have done part a), and I know that
flux of F = divergence x area
but for part b), i m not given a vector field F. so how am I suppose to approach this question and possibly find the divergence (thus the area)? any hint or solution would be much appreciated. ^__^

 

[HallsofIvy]

Since this is in the plane, it is simpler to use Green's theorem rather than the divergence theorem.

Green's theorem says that
$$\oint (Ldx+ Mdy)= \int\int \left(\frac{\partial M}{\partial x}- \frac{\partial L}{\partial y}\right) dA$$

The integral on the right will be the area as long as
$$\frac{\partial M}{\partial x}- \frac{\partial L}{\partial y}= 1$$

One such choice is M(x,y)= x, L(x,y)= 0.