Length of arc in calculus.

Question

We know the formula for length of arc of function f(x) is

L=[INTEGRAL FROM a to b] √ 1+[f'(x)^2] dx.

I know the proof and how it is derived. i know f'(x) is the derivative of f(x).

√ 1+[f'(x)^2] is an entirely different function, isn't it like we end up calculating the length of arc for this function?

Answer 1

Hi Rahul,

Think of it like a formula. You plug in your function f(x) and bounds a and b, then the formula will spit out the arc length. The √(1+[f'(x)^2]) is a consequence of the proof of the formula, and you should write it with the dx as well.

Technically, the formula for ds (s is arc length) is ds = √(dx² + dy²), then take out dx from the square root by dividing through to get ds = dx√((dx² + dy²) / dx²) = dx*√(1+ dy²/dx²), which is nothing but ds = √(1+[f'(x)^2]) dx. This is the formula for the differential of the arc length, and to find the arc length, you must integrate both sides. This is why√(1+[f'(x)^2]) dx is just part of the formula, and not calculating the arc length of the function √(1+[f'(x)^2]) itself.

Hope this helps! Please let me know if you have any questions.

Akshat Y.