Question

Find the length of the curve. r(t) = 3 i + t2 j + t3 k, 0 ≤ t ≤ 1

Answer 1

Call the curve $C$, then the length of $C$ is given by the line integral

$\displaystyle\int_C\mathrm dS=\int_{t=0}^{t=1}\|\mathbf r'(t)\|\,\mathrm dt=\int_{t=0}^{t=1}t\sqrt{9t^2+4}\,\mathrm dt=\frac{13\sqrt{13}-8}{27}$