Problems

01

Arc length - reversed $x$ and $y$ roles

Find the arc length of the curve that satisfies the equation $x=\frac{1}{12}{y}^3+y^{-1}$ over $y\in [\mathrm{1,2}]$.

02

Arc length - tricky algebra

Find the arc length of the curve $y=\frac{x^4}{8}+\frac{1}{4x^2}$ for $x\in [\mathrm{1,2}]$.

(Hint: expand under the root, then simplify, then factor; now it’s a square and the root disappears.)

03

Arc length - tricky integration

Find the arc length of the curve $y=e^x$ for $x\in [\mathrm{0,1}/2]$.

(Hint: the integral can be done using either: (i) $u$-sub then trig sub, or (ii) ‘rationalization’ then partial fractions.)