# Thread: Area enclosed by a complex curve

1. ## Area enclosed by a complex curve

What is the simplest way to find the area enclosed by a curve in the complex plane. I know 2 methods: one based on green's theorem and the other on the substitution rule of integration. Does anyone know any other methods? I'm in a hurry right now so if you need more information, just ask. Thanks to anyone who replies. Happy Thursday (from Seattle). !!

2. what exactly do you mean by complex curve?
since you mentioned green's theorem i'm assuming you're taking vector calculus.

vector calculus doesn't have much... Just green's theorem, strokes theorem, and guass's theorem (aka divergence theorem).

3. ## clarification

By a complex curve I mean a function that maps a value z in the complex plane to another value w in the plane or two real value functions x(t) + i*y(t) that produce a curve like w=1/z when |z|=1 which is equal to cos(t) - i*sin(t). I mention green's theorem because from it a formula for the region is derived for the area under a parametric curve.
$\displaystyle A=\frac{1}{2}\int_a^b{(xy'-yx') \:dt}$
The other derived from the substitution rule also works
$\displaystyle A=\int_a^b{(yx') \:dt}$, though the interval is not always [a,b] but sometimes [b,a]. I was wondering if these were the only methods out there, especially any that didn't require spliting into real and imaginary functions. The function I'm using is in w=f(z) form and would be easier to deal with in this form. Thanks for your reply.

4. i believe green's theorem is simplest for solving such problems as far as i know. Just becareful not to have singularities in your line integral as green's theorem may be invalid in such cases.