How to use Green's theorem on unit circle

Question

Compute $∫_{\gamma} −y^3 dx + x^3 dy$, where $\gamma$ is the positively oriented edge of the unit circle.

So I used Green's theorem and got $$\begin{align}∫_{\gamma} −y^3 dx + x^3 dy &=\iint_{D} 3x^2 + 3y^2 dxdy\\ &=\iint_{D} 3r^2\cos^2(\theta) + 3r^2\sin^2(\theta) dr d\theta\\ &=\int_0^{2\pi} d\theta\int_0^1 3r^2 dr = 2\pi. \end{align}$$

But the correct answer is $\frac{3\pi}2$.

Answer 1

The application of Green's theorem is correct, but $$\iint_D 3x^2+3y^2 \,\mathrm dx\,\mathrm dy \ne \iint_D 3(r\cos \theta)^2 + 3(r\sin\theta)^2 \,\mathrm dr\,\mathrm d\theta.$$ Replace $\mathrm dx\,\mathrm dy$ by $r\,\mathrm dr\,\mathrm d\theta$ every time you use polar coordinates.