## Integral Calculus, The Area of Part of a Circle

### The Area of Part of a Circle

Now that we know about trig functions, let's compute the area of the circle again, this time using integration.  This lets us compute the area of part of a circle, the portion on either side of a chord.

The area bounded between the upper half of the unit circle and the x axis, from 0 to x, is the integral of sqrt(1-t2) as t runs from 0 to x.  Replace t with sin(θ), giving an integrand of cos(θ)2.  By the double angle formula, this is cos(2θ)/2 + 1/2.  Integrate to get sin(2θ)/4+θ/2.  Use the double angle formula again to get (sin(θ)cos(θ) + θ) over 2.  To cover the quarter circle, let t run from 0 to 1, hence θ runs from 0 to π/2.  This gives an area of π/4, hence the area of the circle is π.

Replace θ with the arcsine of x, remembering that the cosine of the arcsine of x is sqrt(1-x2).  The area from 0 to x is now x×sqrt(1-x2) + asin(x) over 2.  As an exercise, differentiate this to get sqrt(1-x2).  When x = 1/2, the area is sqrt(3)/8 + π/12, approximately 0.4783.  This is roughly 61% of the quarter circle, leaving the other 39% to the right of the chord x = 1/2.