Chapter 6.4, Problem 193E

In the following exercises, use the binomial approximation

$\sqrt{1-x}\approx 1-\frac{x}{2}-\frac{x^2}{8}-\frac{x^3}{16}-\frac{5x^5}{128}-\frac{7x^5}{256}$ for |x| <1 to approximate each number. Compare this value to the value given by a scientific calculator.

193, [T] Recall that the graph of $\sqrt{1-x^2}$ is an upper semicircle of radius 1. Integrate the binomial approximation of $\sqrt{1-x^2}$ up ro order 8 from x = —1 to x = 1 to estimate $\frac{\pi }{2}$ .