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.
30. ∫(0 to 6) (x³/16 - x) dx = 4

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with approximating the integral ∫(0 to 6) (x³/16 - x) dx using the Midpoint Rule and the Trapezoid Rule for different values of n (number of subintervals). Additionally, you will compare the approximations to the exact value of the integral, which is given as 4, and compute the errors.
2. Step 2: Define the integral and the interval. The function to integrate is f(x) = (x³/16 - x), and the interval of integration is [0, 6]. The exact value of the integral is provided as 4.
3. Step 3: For the Midpoint Rule, divide the interval [0, 6] into n subintervals of equal width Δx = (6 - 0)/n. For each subinterval, calculate the midpoint xᵢ = (xᵢ₋₁ + xᵢ)/2, where xᵢ₋₁ and xᵢ are the endpoints of the subinterval. Evaluate f(xᵢ) at each midpoint and sum up the contributions Δx * f(xᵢ) for all subintervals.
4. Step 4: For the Trapezoid Rule, divide the interval [0, 6] into n subintervals of equal width Δx = (6 - 0)/n. For each subinterval, calculate the average of the function values at the endpoints, f(xᵢ₋₁) and f(xᵢ). Sum up the contributions Δx * (f(xᵢ₋₁) + f(xᵢ))/2 for all subintervals.
5. Step 5: Create a table to compare the results. For each value of n (4, 8, 16, 32), compute the approximations using both the Midpoint Rule and the Trapezoid Rule. Calculate the error for each approximation by subtracting the exact value (4) from the approximation. Organize the results in a table similar to Table 8.5, showing n, the Midpoint Rule approximation, the Trapezoid Rule approximation, and their respective errors.

Key Concepts

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.

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.

Error Analysis in Numerical Integration

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