Find the length of the arc of the semi-cubical parabola y² = x³ between the points (1, 1) and (4, 8).

Solution:

Given,

y² = x³

y = x^3/2

\(\frac{dy}{dx} = \frac{3}{2} x^{\frac{1}{2}}\)

The arc length formula gives

L = \(\int_1^4 \sqrt{1 + \left ( \frac{dy}{dx} \right )^2 dx}\)

= \(\int_1^4 \sqrt{1 + \frac{9}{4}x dx}\)

If we substitute

u = \(1 + \frac{9}{4}x\), then \(du = \frac{9}{4} dx\)

when x = 4 then u  = 10

when x = 1 then u = 13/4

Therefore

L = \(\frac{4}{9} \int_{\frac{13}{4}}^{10} \sqrt{u} \enspace du\)

= \(\left [ \frac{4}{9} . \frac{2}{3} u^{\frac{3}{2}} \right ]_{\frac{13}{3}}^{10}\)

= \(\frac{8}{27} \left [ 10^{\frac{3}{2}} – \left ( \frac{13}{4} \right )^{\frac{3}{2}} \right ]\)

= \(\frac{1}{27} \left ( 80 \sqrt{10} – 13\sqrt{13} \right )\)