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

Short Answer

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 given curve $C$ .
$$\begin{array} { l } \mathbf { F } ( x , y ) = x ^ { 3 } y ^ { 4 } \mathbf { i } + y ^ { 3 } x ^ { 4 } \mathbf { j } \\C : \mathbf { r } ( t ) = \sqrt { t } \mathbf { i } + \left( 1 + t ^ { 3 } \right) \mathbf { j } , 0 \leq t \leq 1\end{array}$$