Chapter 6.5, Problem 23E

To determine

To calculate: The Simpson’s Rule approximations of the function with given values.

Answer to Problem 23E

The Simpson’s Rule approximations of the function with given n=50 and n=100 is 3.13791 and 3.14029 respectively.

Explanation of Solution

Given information:

  $2\sqrt{1-x^2}$

Calculation:

Simpson's rule is used to find the value of integral.

To approximate ${{\displaystyle \int }}_a^{b}f(x)dx,$ , use

  $S=\frac{h}{3}\left(y_0+4y_1+2y_2+4y_3+\dots .+2y_{n-2}+4y_{n-1}+y_n\right)$ .

Where $[a,b]$ is partitioned into an even number $n$ of subintervals of equal length $h=\frac{b-a}{n}$ To Find integral, use calculator program of Simpson's rule approximations with-

  $f(x)=2\sqrt{1-x^2}$

  $\begin{array}{l}\left[a,b\right]=\left[–1,1\right]\hfill \\ n=50\hfill \\ n=100\hfill \end{array}$

NA

function	n	Value of integral
$2\sqrt{1-x^2}$	50	3.13791
$2\sqrt{1-x^2}$	100	3.14029