Question

Evaluate the line integral f · dr, c where c is given by the vector function r(t). f(x, y, z) = x i + y j + xy k, r(t) = sin(t) i + cos(t) j + t k, 0 ≤ t ≤ π

Answer 1

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=\pi}\mathbf f(\mathbf r(t))\cdot\mathrm d(\sin t\,\mathbf i+\cos t\,\mathbf j+t\,\mathbf k)$
$=\displaystyle\int_0^\pi\bigg(\sin t\,\mathbf i+\cos t\,\mathbf j+\sin t\cos t\,\mathbf k\bigg)\cdot\bigg(\cos t\,\mathbf i-\sin t\,\mathbf j+\mathbf k\bigg)\,\mathrm dt$
$=\displaystyle\int_0^\pi\sin t\cos t\,\mathrm dt=0$