Question

What is the arc length of `f(x)=sqrt(sinx)` in the interval `[0,pi]`?

Answer 1

The formula for arc length on interval `[a, b]` is

`A = int_a^b sqrt(1 + (f'x)^2) dx`

The derivative of `f(x)` will be obtained using the chain rule.

`f'(x) = cosx * 1/(2sqrt(sinx))`

`f'(x) = cosx/(2sqrt(sinx)`

Using the given formula:

`A = int_0^pi sqrt(1 + (cosx/(2sqrt(sinx)))^2) dx`

`A = int_0^pi sqrt(1 + cos^2x/(4sinx)) dx`

`A = int_0^pi sqrt(1 + 1/4cotxcosx) dx`

Which according to the integral calculator has no solution through elementary antiderivatives. A numerical approximation for arc length gives

`A= 4.04` units

to three significant figures.

Hopefully this helps!