Let C Be the Closed Path From $( 0,0 )$ To $( 2,4 )$

Short Answer

Let C be the closed path from $( 0,0 )$ to $( 2,4 )$ along $y = x ^ { 2 }$ and back again from $( 2,4 )$ to $( 0,0 )$ along $y = 2 x$ . Evaluate $\int _ { C } \left( x ^ { 3 } + 2 y \right) d x + \left( x - y ^ { 2 } \right) d y$ directly, and then using Green's Theorem.