Let $\phi ( x , y ) = 3 x ^ { 2 } y ^ { 3 }$

Short Answer

Let $\phi ( x , y ) = 3 x ^ { 2 } y ^ { 3 }$ . Note that $\nabla \phi = \left\langle 6 x y ^ { 3 } , 9 x ^ { 2 } y ^ { 2 } \right\rangle$ . Evaluate the line integral $\int _ { C } 6 x y ^ { 3 } d x + 9 x ^ { 2 } y ^ { 2 } d y$ where C is the curve $x = t e ^ { t ^ { 3 } - 1 } , \mathrm { y } = ( 2 t + 1 ) \cos ( 2 \pi t )$ , $0 \leq t \leq 1$ .