How to calculate the arc length of a implicite curve like

Question

I have to solve this:

Calculate the arc length of the curve $(\frac xa)^\frac 2 3 + (\frac xb)^\frac 2 3= 1$ from x=$\frac a8$ to $x=a$

The problem is that the equation has just one variable so I do not know how to get a function $f(x)$ or $g(y)$ to apply the formula $L= \int\sqrt{(1+(\frac{dy}{dx})^2)} dx$.

Any ideas?

Answer 1

Let put

$$(\frac{x}{a})^{\frac{1}{3}}=\cos(t)$$

or

$$x=a\cos^3(t)$$

and in the same way

$$y=b\sin^3(t)$$

thus

$$dx=-3a\cos^2(t)\sin(t)dt$$

and

$$dy=3b\sin^2(t)\cos(t)dt$$

which give

$$dl=\sqrt{(dx)^2+(dy)^2}=3|\sin(t)\cos(t)|\sqrt{a^2\cos^2(t)+b^2\sin^2(t)}$$

and we integrate from $t=\frac{\pi}{3}$ which corresponds to $x=\frac{a}{8}$ to $t=2\pi$ corresponding to $x=a$, by putting $u=\sin^2(t)$ .