Chapter 6.4, Problem 248E

The following exercises deal with Fresnel integrals.

248. [T] Use Newton’s approximation of the binomial $\sqrt{1-x^2}$ to approximate $\pi$ as follows. The circle centered at $\left(\frac{1}{2},0\right)$ with radius $\frac{1}{2}$ has upper semicircle $y=\sqrt{x}\sqrt{1-x}$. The sector of this circle bounded by the x-axis between x = 0 and $x=\frac{1}{2}$ and by the line joining $\left(\frac{1}{4},\frac{\sqrt{3}}{4}\right)$ , corresponds to $\frac{1}{6}$ of the circle and has area $\frac{\pi }{24}$ . This sector is the union of a right triangle with height $\frac{\sqrt{3}}{4}$ and base $\frac{1}{4}$ and the region below the graph between x = 0 and $x=\frac{1}{4}$. To find the area of this region you can write $y=\sqrt{x}\sqrt{1-x}=\sqrt{x}\times$ (binomial expansion of $\sqrt{1-x}$ ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate $\pi$ .