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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