Use Green's Theorem to evaluate this problem? Step by step solution?

Question

Use Green's Theorem to evaluate this problem? Step by step solution?

For a vector field $$F(x,y)=\frac{y^2}{(1+x^2)}i+2y\arctan(x)j$$ find a function f such that $F(x,y)=∇f$ and use this result to evaluate $\int_CF\cdot dr$ where $C: r(t)=t^2i+2(t)j;~~ 0≤t≤1$.

sorry guys I'm new to the coding. But I tagged the word problem if you click on the title. Thanks to Razieh for the coding help.

Answer 1

Based on their instructions, they want you to use the fundamental theorem of calculus for line integrals. Green's theorem only applies to a closed, simple path for which the field is continuously differentiable inside the curve.

It should be obvious that $f(x,y)=y^2\arctan(x)$, and so by the FTCLI, we have $$\int_CF\cdot dr=f(1,2)-f(0,0)=4\arctan(1)-0=\pi.$$