How to find the length of a curve using calculus

Integrate:

sqrt(1 + (dy/dx)^2) * dx


y = cosh(x)
dy/dx = sinh(x)
(dy/dx)^2 = sinh(x)^2

sqrt(1 + sinh(x)^2) * dx =>
sqrt(cosh(x)^2) * dx =>
cosh(x) * dx

Integrate

sinh(x) + C

From 0 to a

sinh(a) - sinh(0) =>
sinh(a) - 0 =>
sinh(a)

Integral Sqrt[1+ (dy/ dx)^2] dx from a to b


Integral (sqrt[1 + sinh[x]^2]dx from 0 to a

= Sinh [a]

ds^2 = dy^2 +dx^2

s = [Int (0 ,a)] [1+(dy/dx)^2}^1/2 dx

y = coshx sub and crank ir out!