Question

Numerical Approximation Use the Midpoint Rule and the Trapezoidal Rule with \(n=4\) to approximate \(\pi\) where \(\pi=\int_{0}^{1} \frac{4}{1+x^{2}} d x\) Then use a graphing utility to evaluate the definite integral. Compare all of your results.

Short answer

The Midpoint Rule and the Trapezoidal Rule are widely used numerical approximation methods for definite integrals. Although they provide reasonable approximations of \(\pi\), the values differ due to the inherent inaccuracies in these methods. The most accurate value of \(\pi\) is obtained by directly evaluating the definite integral using a graphing utility.

Step-by-step solution

Dividing the Interval

As \(n=4\), this means that the interval from 0 to 1 is divided into 4 equal parts. These are \([0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1]\). Midpoints of these intervals are needed for the Midpoint Rule, i.e., \(0.125, 0.375, 0.625, 0.875\). The endpoints are needed for the Trapezoidal Rule.

Using the Midpoint Rule

For the Midpoint Rule, plug midpoints into the function and add these values up. The Midpoint Rule is approximately \(h[f(x_1)+f(x_2)+f(x_3)+f(x_4)]\) where \(h=0.25\), the length of the subintervals, and \(x_1,x_2,x_3,x_4\) are midpoints of the subintervals. After substituting and solving, the result will be the approximate value of \(\pi\) using the Midpoint Rule.

Using the Trapezoidal Rule

The Trapezoidal Rule states that the integral is approximately equal to \(\frac{h}{2}[f(a) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(b)]\), where \(a\) and \(b\) are the endpoints of the interval and \(h\) is the length of the subintervals. Applying this formula and calculating will give the approximate value of \(\pi\) using the Trapezoidal Rule.

Evaluating the Definite Integral with a Graphing Utility

A graphing utility or a calculator with inbuilt integration functionality can be used to evaluate the integral to get the exact value of \(\pi\). Just input the function \(\frac{4}{1+x^2}\) and the limits of integration (0 to 1). This direct method will provide a more accurate result of \(\pi\) for comparison purposes.

Key concepts

Midpoint Rule

The Midpoint Rule is a numerical technique used to approximate the value of a definite integral. This method involves dividing the integral's interval into subintervals, calculating the function's value at the midpoint of each subinterval, and then using these values to estimate the area under the curve.

• Each subinterval should have equal width, denoted by \( h \).
• The midpoints are the average of the endpoints of each subinterval. For example, between 0 and 0.25, the midpoint is 0.125.
• The formula used for the Midpoint Rule is: \( h[f(x_1) + f(x_2) + \, \ldots \, + f(x_n)] \).

To approximate \( \pi \) using the Midpoint Rule, we divide the interval [0, 1] into four subintervals, each of width 0.25. The midpoints are 0.125, 0.375, 0.625, and 0.875.
You substitute these midpoints into the function \( \frac{4}{1+x^2} \), sum the results, then multiply by the width of the subintervals (0.25). This gives an approximation of the integral, and therefore, approximately \( \pi \).

Trapezoidal Rule

The Trapezoidal Rule is another numerical method used to approximate definite integrals. It works by approximating the region under a curve as a series of trapezoids rather than rectangles.

• Like the Midpoint Rule, it uses equally spaced subintervals.
• The endpoints of these subintervals become the corners of the trapezoids.
• The formula is a bit different: \( \frac{h}{2}[f(a) + 2f(x_1) + 2f(x_2) + \, \ldots \, + f(b)] \), where \( a \) and \( b \) are the endpoints of the interval.

Using the Trapezoidal Rule on the same interval [0, 1], with subintervals of width 0.25, involves calculating each function value at the endpoints: 0, 0.25, 0.5, 0.75, and 1.
Then, the function values at the intermediate points are doubled, you sum all the terms, and multiply by \( \frac{h}{2} \). This calculation approximates the integral and gives you another estimate of \( \pi \).

Definite Integral

A definite integral is a fundamental concept in calculus that represents the exact area under a curve between two limits. Unlike numerical approximations, evaluating a definite integral gives an exact result if the integral is solvable analytically.

• It is represented as \( \int_{a}^{b} f(x) \, dx \).
• The result is a number that signifies the net area observed on the graph from \( x = a \) to \( x = b \).

In the exercise, we are focusing on the definite integral of \( \frac{4}{1+x^2} \) from 0 to 1, which gives \( \pi \).
A graphing calculator or utility can be used to evaluate this integral directly for increased accuracy. By computing this integral precisely, you achieve a result to compare against the approximate values found using the Midpoint and Trapezoidal Rules, helping highlight the accuracy and potential error in numerical methods.