Question

Find the length of the arc along f(x) = integral from 0 to x^3 sqrt(cos t) dt on the set of x [0, pi/3].

Answer 1

You need to integrate sqrt[1+f'(x)^2] from x = 0 to pi/3. Computing the derivative of f(x) is not difficult, you can use the chain rule, substitute u = x^3 for the upper limit and use that the derivative w.r.t. x is the derivative w.r.t u times the derivative of of u w.r.t. x. The derivative w.r.t. u is, by the Fundamental Theorem of Calculus, equal to sqrt[cos(u)].