How to find the area bounded by three or more curves

Question

The area bounded by two curves can found by subtracting the integrands.Is there a general way to find the area bounded by three or more curves?

for instance:

${ y }^{ 3 }-{ x }^{ 3 }-x+xy=0$,$x-{y}^3-{x}^2-y=0$, $xy$=sin$(x-y)$

Answer 1

One way that will work for any number of curves $\gamma_1(t), \gamma_2(t), \ldots$ is to use Stokes's theorem:

$$A = \int_{\Omega} 1\,dA = \int_{\Omega} \frac{1}{2}\nabla \cdot (x,y)\,dA = \frac{1}{2}\int_{\partial \Omega} (x,y)\cdot \hat{n}\,ds = \frac{1}{2}\sum_{i=1}^n \int_0^1 \gamma_i \times \gamma_i'\,dt,$$

where the term inside the integral on the right is the two-dimensional cross product $(a,b) \times (c,d) = ad-bc.$

Notice that this assumes you orient your curves counterclockwise, and calculates the signed area enclosed -- so if your curves intersect themselves or each other (e.g. a figure-eight) you will need to split the calculation into separate pieces.