The area of the segment dA is L da. We know by the Pythagorean theorem that the length of L is twice the length sqrt(r^2-a^2). Thus:
To find the summation the total area of the circle, we will sum up all of the dA’s from a=−r to a=r:
This being a complex integral, the use of trigonometric substitution is necessary for solving it. In figure 2, the triangle is shown in more detail. The value for a could be replaced by a trigonometric equality, a=r sin θ.The integral therefore becomes:
Making the final substitutions:
(cos θ)^2=(1 + cos 2θ)/2. We then derive:
And there you have it, the area of a circle. You might have thought it would be a little more simple than that, but you were wrong.