Stokes’s theorem and compact 2-manifolds in $\mathbb{R}^2$

Question

I am trying to solve the following problem of Andrew Browder: "Mathematical Analysis; An Introduction" (Springer Undergraduate Texts in Mathematics):

Find THE compact $2$-manifold $M$ in $\mathbb{R}^2$ with area $\pi$ for which $$ \int_{\partial M} {y^3dx + (3x - x^3)dy} $$ is maximal.

The definition of Manifold in the book is:

Definition

I have tried to use the Stokes’s theorem but I don't know how to find a such manifold $M$. Thanks in advance.

Answer 1

Stokes' theorem gives you

$$\int_{\partial M}y^3dx+(3x-x^3)dy=\int_Md(y^3dx+(3x-x^3)dy)=3\int_M(1-(x^2+y^2))dxdy.$$

Since the integrand is positive iff $x^2+y^2\leq 1$, it appears that the integral will be maximal on $M=B(0,1)$ the ball of center $0$ and radius $1$.