Question

In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{-1}^{1} 2 \sqrt{1-x^{2}} d x$$

Short answer

To solve this definite integral approximation using Simpson’s Rule, we just have to follow the step by step instructions previously explained for each \(n\). The actual numerical results will depend on the calculator or program you are using.

Step-by-step solution

Define Simpson's Rule

Simpson's Rule is a method of numerical integration which is a way to approximate the definite integral of a function. According to Simpson's Rule, the integral of a function can be estimated by the sum: \[\frac{b-a}{6n}[f(a) + 4\sum_{i=1}^{n}f(a + h(2i - 1)) + 2\sum_{i=1}^{n-1}f(a + 2hi) + f(b)]\] where \(a\) and \(b\) are the lower and upper limits of integration, respectively, \(f(a)\) is the value of the function at \(a\), \(f(b)\) is the value of the function at \(b\), \(n\) is the number of subintervals, and \(h\) is the width of each subinterval, calculated as \((b-a)/n\).

Set up the Integral with Simpson's Rule for \(n=50\)

First, set up the integral as dictated by Simpson's Rule, where our \(f(x) = 2\sqrt{1 - x^2}\), \(a=-1\), \(b=1\), and \(n=50\). Calculate \(h\) as \((b-a)/n\). Once that's set, calculate each term of the sum, plug them into Simpson's Rule, and compute the entire sum. If you use a calculator or programming, this step should be fairly straightforward.

Set up the Integral with Simpson's Rule for \(n=100\)

Repeat the above steps, but this time use \(n=100\) instead of \(n=50\). Again, calculate each term of the sum, plug them into Simpson's Rule, and compute the entire sum.