Calculate $\int_{\rm C}(e^{x^2}- y{x}^{2}){\rm d}x+ \cos({y}^{2}){\rm d}y$

Question

I am trying to calculate the curve integral $$\int_{\rm C}(e^{x^2}- y{x}^{2}){\rm d}x+ \cos({y}^{2}){\rm d}y$$ where ${\rm C}$ is the unit circle passed through one lap counter clockwise. I am using Green's theorem which renders me the new integral $$\iint x^2{\rm d}x{\rm d}y$$ and using polar coordinates I get the integral $$\iint r^3\cos^2\theta{\rm d}\theta{\rm d}r$$ where I integrate $\theta$ from $0$ to $2\pi$ and $r$ from $0$ to $1$. This gives me $0$ as the result but that is not the correct answer. What have I done wrong in my calculations ?

Answer 1

Everything else is fine but you have a mistake in your integral.

$\displaystyle \iint_R r^3 cos^2\theta \ d\theta \ dr = \int_0^1 \int_0^{2\pi} r^3 \frac{1+cos2\theta}{2} \ d\theta \ dr$

$ = \displaystyle \int r^3 \bigg[\frac{2\theta+ \sin2\theta}{4}\bigg]_0^{2\pi} \ dr = \int_0^1 \pi r^3 dr = \frac{\pi}{4}$