Finding a line integral

Question

I am asked the following question:

For F=yx^3i+y^2j, find the line integral F*dr on the portion of the curve y=x^2 from (0,0) to (1,1).

The answer is 1/2. I am attempting to use Green's Theorem. My bounds are: 0<x<1 and x^2<y<1. I am getting -1/12 repeatedly. How do I solve this problem using Green's Theorem?

Answer 1

∫_C(yx³dx + y²dy) = ∫₀¹(x⁵ + 2x⁵)dx = x⁶/2₀¹ = 1/2

Answer 2

You can not use Green's Theorem for this problem. The reason is, one of the prerequisites

of the theorem is that the curve must be a simply closed one.