Evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$

Evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ for the vector field F and the path C. (Hint: Show that F is conservative, and pick a simpler path.) $\mathbf { F } ( x , y ) = \left( 16 x ^ { 3 } y ^ { 2 } - 4 y \sin x \right) \mathbf { i } + \left( 8 x ^ { 4 } y + 4 \cos x \right) \mathbf { j }$ C: $\mathbf { r } ( t ) = ( 3 - \sin t ) \mathbf { i } + 2 \cos t \mathbf { j } ; \quad 0 \leq t \leq \pi$

A) $- 16 \cos ( 3 )$
B) 54
C) 0
D) 19

Correct Answer:

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