Question

Evaluate the line integral c f · dr, where c is given by the vector function r(t). f(x, y) = xy i + 6y2 j r(t) = 16t6 i + t4 j, 0 ≤ t ≤ 1

Answer 1

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}\mathbf f(\mathbf r(t))\cdot\mathrm d(16t^6\,\mathbf i+t^4\,\mathbf j)$
$=\displaystyle\int_0^1\bigg(16t^{10}\,\mathbf i+6t^8\bigg)\cdot\bigg(96t^5\,\mathbf i+4t^3\,\mathbf j\bigg)\,\mathrm dt$
$=\displaystyle\int_0^1(1536t^{15}+24t^{11})\,\mathrm dt=98$