What is the arc length of #f(x)=cosx# on #x in [0,pi]#?
Arc length of a curve
This formula is derived by adding up infinitesimal arc lengths along the curve. Here, for example, is an online explanation: http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx
In this case
Now here we have run straight into the problem I warned about above! This is not an integral that is solvable with elementary functions. Indeed, it is an integral that is part of a famous problem from mathematical history - the calculation of the arc length of the ellipse. The solution of this equation is a transcendental function, one that is simply defined to solve this particular equation, the (complete) Elliptic Integral of the Second Kind:
This is defined as
which now matches our limits of integration.
Numerically, this is approximately 3.82.