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**manyarrows** Given problem: Evaluate the line integral $\displaystyle \int_C \mathbf {F}\cdot d\mathbf r$, where C is given by the vector function $\displaystyle \mathbf r(t)$.

$\displaystyle \mathbf F(x,y,z)=\sin x\mathbf i+\cos y\mathbf j+xz\mathbf k$

$\displaystyle \mathbf r(t)=t^3\mathbf i-t^2\mathbf j+t\mathbf k$

$\displaystyle 0\leq t\leq 1$

Beagle work:

$\displaystyle x=t^3, y=t^2, z=t$

$\displaystyle \int^1_0 [\sin (t^3)\mathbf i+\cos(-t^2)\mathbf j+(t^3)(t)\mathbf k]\cdot[3t^2\mathbf i-2t\mathbf j+\mathbf k]dt$

$\displaystyle \int^1_0 \left[3t^2\sin(t^3)-2t\cos(-t^2)+t^4\right] dt$

Thank the beagle