Question

Find the length of the arc formed by x2=10y3 from point a to point b, where a=(0,0) and b=(100,10).

Answer 1

We can parameterize the curve by setting $y=t$, then

$x^2=10t^3\implies x=\sqrt{10}\,t^{3/2}$

The arc length of $x^2=10y^3$, parameterized by $\mathbf r(t)=(x(t),y(t))=(\sqrt{10}\,t^{3/2},t)$, is given by the integral

$\displaystyle\int_0^{10}\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt=\int_0^{10}\sqrt{\frac{45}2t+1}\,\mathrm dt$

$=\displaystyle\frac2{45}\int_1^{226}\sqrt u\,\mathrm du$ where $u=\dfrac{45}2t+1$

$=\dfrac4{135}u^{3/2}\bigg|_1^{226}=\dfrac{4(226\sqrt{226}-1)}{135}\approx100.638$