# What is the arclength of f(t) = (t^3-t^2+5t,9t) on t in [1,4]?

Use $L = {\int}_{1}^{4} \left(\sqrt{{\left(\frac{d \left(x \left(t\right)\right)}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy} \left(t\right)}{\mathrm{dt}}\right)}^{2}}\right) \mathrm{dt}$ where $x \left(t\right) = {t}^{3} - {t}^{2} + 5 t$ and $y \left(t\right) = 9 t$
$L = {\int}_{1}^{4} \left(\sqrt{{\left(3 {t}^{2} - 2 t + 5\right)}^{2} + {9}^{2}}\right) \mathrm{dt}$
$L \approx 70.05$