# The long and winding road

Calculus Level 4

We have a vector field $$\vec{F} : \mathbb{R}^2 \to \mathbb{R}^2$$ such that $$\vec{F}(x,y)=(2x \sin y,x^2 \cos y)$$. Let:

• $$C_1$$ be the segment of $$y=x^3$$ from $$(0,0)$$ to $$(1,1)$$
• $$C_2$$ be the line segment from $$(1,1)$$ to $$(2,4)$$
• $$C_3$$ be the line segment from $$(2,4)$$ to $$\left(\pi,\frac{\pi}{2}\right)$$
• $$C=C_1 \cup C_2 \cup C_3$$

Find $$\displaystyle \int\limits_C \vec{F} \cdot \mathrm{d}\vec{r}$$ to four decimal places.

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