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

##### 1 Answer

Jan 8, 2018

The formula for arc length on interval

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

The derivative of

#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!