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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.8.33
Chapter 8, Problem 8.8.33

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
33. ∫(0 to π) sin x cos(3x) dx = 0

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Step 1: Understand the problem. You are tasked with approximating the integral ∫(0 to π) sin(x)cos(3x) dx using the Midpoint Rule and the Trapezoid Rule for different values of n (number of subintervals). Additionally, you need to compute the errors by comparing the approximations to the exact value of the integral, which is given as 0.
Step 2: Divide the interval [0, π] into n subintervals. For each value of n (4, 8, 16, 32), calculate the width of each subinterval, denoted as Δx, using the formula Δx = (b - a) / n, where a = 0 and b = π.
Step 3: Apply the Midpoint Rule. For each subinterval, find the midpoint, evaluate the function sin(x)cos(3x) at the midpoint, and sum up the contributions from all subintervals. Multiply the sum by Δx to approximate the integral.
Step 4: Apply the Trapezoid Rule. For each subinterval, evaluate the function sin(x)cos(3x) at the endpoints of the subinterval, average the two values, and sum up the contributions from all subintervals. Multiply the sum by Δx to approximate the integral.
Step 5: Compute the error for each approximation. Subtract the exact value of the integral (0) from the approximations obtained using the Midpoint Rule and the Trapezoid Rule. Create a table showing the approximations and errors for n = 4, 8, 16, and 32.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Midpoint Rule

The Midpoint Rule is a numerical method for approximating the definite integral of a function. It involves dividing the interval into subintervals, calculating the function's value at the midpoint of each subinterval, and then summing these values multiplied by the width of the subintervals. This method tends to provide better approximations than using the endpoints, especially for functions that are relatively smooth.
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Trapezoid Rule

The Trapezoid Rule is another numerical integration technique that approximates the area under a curve by dividing the interval into subintervals and forming trapezoids. The area of each trapezoid is calculated using the average of the function values at the endpoints of each subinterval, multiplied by the width of the subinterval. This method is generally more accurate than using rectangles, particularly for linear functions.
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Error Analysis in Numerical Integration

Error analysis in numerical integration involves assessing the difference between the exact value of an integral and its numerical approximation. The error can be influenced by factors such as the number of subintervals (n) used in the approximation and the behavior of the function being integrated. Understanding how to compute and interpret this error is crucial for evaluating the effectiveness of numerical methods like the Midpoint and Trapezoid Rules.
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