Question

Determine the length of curve 𝑥 =

2/3

(𝑦 − 1)raise to 3/2 when 𝑦 ∈ [1,4]

Answer 1

$\displaystyle x = \frac{2}{3} \left(y - 1\right)^{\frac{3}{2}}\\ \frac{\mathrm{d}x}{\mathrm{d}y} = \left(y - 1\right)^{\frac{1}{2}}\\ \left(\frac{\mathrm{d}x}{\mathrm{d}y}\right)^2 = y - 1\\ 1 + \left(\frac{\mathrm{d}x}{\mathrm{d}y}\right)^2 = y\\ \sqrt{1 + \left(\frac{\mathrm{d}x}{\mathrm{d}y}\right)^2} = \sqrt{y}\\ \begin{aligned} \int_1^4\sqrt{1 + \left(\frac{\mathrm{d}x}{\mathrm{d}y}\right)^2}\,\mathrm{d}y &= \int_1^4 \,\,\sqrt{y}\,\mathrm{d}y \\&= \frac{2y^{\frac{3}{2}}}{3}\biggr\vert_1^4 = \frac{16 - 2}{3} = \frac{14}{3} \end{aligned}$