Question

find the length of the arc of the parabola y^2=4ax

Answer 1

Heyy!
Here is your answer

Arc lenght :

1) S = 1/2a(4ax+1)^3/2. / 3

This is the required arc lenght.

For calculation , see pic

Answer 2

${y}^{2} = 4ax$

$\implies f(x) = y = \pm2 \sqrt{ax}$

As this is an Even Function w.r.t y :

$\\ S = 2 \int_{0} ^{ x_{0}} \sqrt{1 + {(f}^{ \: ’}(x)) ^{2} } \: dx$

$\\ S = 2 \:\int_{0} ^{ x_{0}} \sqrt{1 + \frac{a}{x} } \: dx$

$\\ S = 2 \sqrt{ax _{0} + {x _{0}}^{2} } + 2 a \ln( \frac{ \sqrt{ {x _{0}} } + \sqrt{ x _{0} + a }}{ \sqrt{a} } )$