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

Evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ for the vector field $\mathbf { F }$ and the path $C$ . (Hint: Show that $\mathbf { F }$ is conservative, and pick a simpler path.)
$$\mathbf { F } ( x , y ) = \left( 18 x ^ { 2 } y ^ { 2 } - 5 y \sin x \right) \mathbf { i } + \left( 12 x ^ { 3 } y + 5 \cos x \right) \mathbf { j }$$
$C : \mathbf { r } ( t ) = ( - 2 - \sin t ) \mathbf { i } + 2 \cos t \mathbf { j } ; \quad 0 \leq t \leq \pi$

A) 24
B) 47
C) 0
D) $- 20 \cos ( - 2 )$

Correct Answer:

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