Find a Function F Such That $\mathbf { F } = \nabla f$

question 171

Essay

Find a function f such that $\mathbf { F } = \nabla f$ and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the curve C. $\mathbf { F } ( x , y ) = y \mathbf { i } + x \mathbf { j };$ C is the arc of the curve $y = x ^ { 4 } - x ^ { 3 }$ from $( 1,0 )$ to $( 2,8 )$ .