The notes contained in the link directly below show the derivation of circle area using some of the integration techniques illustrated earlier; the formula for arc length is also derived, with reference to the Mean Value Theorem. This formula is applied to several functions to determine arc length over a given interval and is ultimately used to prove the formula for circumference of a circle. This circumference formula is then used to once again prove the formula for area of circles, this time using the "shell" method of integration.
Circle Area & Arc Length - Samuelson
The links below reinforce the ideas presented in my notes and provide much more information on those topics.
Arc Length - Riemann Sum
Mean Value Theorem - Interactive
Arc Length, Area, and the Arcsine Function