Question

Find the line integral of f⃗ (x,y,z)=x3i⃗ +y2j⃗ +zk⃗ along the line segment c from the origin to the point (2,3,9).

Answer 1

The path $\mathcal C$ can be parameterized by the vector-valued function,

$\mathbf r(t)=t(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)$

where $0\le t\le1$. Then

$\mathrm d\mathbf r=(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)\,\mathrm dt$

The line integral reduces to

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}(x(t)^3\,\mathbf i+y(t)^2\,\mathbf j+z(t)\,\mathbf k)\cdot(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)\,\mathrm dt$

$=\displaystyle\int_{t=0}^{t=1}((2t)^3\,\mathbf i+(3t)^2\,\mathbf j+9t\,\mathbf k)\cdot(2\,\mathbf i+3\,\mathbf j+9\,\mathbf k)\,\mathrm dt$

$=\displaystyle\int_{t=0}^{t=1}(16t^3+27t^2+81t)\,\mathrm dt=\frac{107}2$