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Chapter 7: Problem 3

Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

Short Answer

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Question: Explain geometrically how the Trapezoid Rule is used to approximate a definite integral and provide a step-by-step method to approximate the definite integral using the trapezoid rule. Answer: The Trapezoid Rule is used to approximate a definite integral by finding the sum of the areas of trapezoids over a specified interval [a, b]. Geometrically, it calculates the signed area under a curve (function f(x)) by dividing the interval [a, b] into n equal subintervals, each with width h, and using trapezoids to approximate the areas. The step-by-step method to approximate the definite integral using the trapezoid rule is as follows: 1. Determine the width h of each subinterval: h = (b - a) / n 2. Calculate the area of each trapezoid: Area = (1/2) * (f(x_i) + f(x_(i + 1))) * h 3. Sum the areas of all trapezoids: Approximate integral = 鈭慱{i = 0}^{n - 1} [(1/2)(f(x_i) + f(x_(i + 1))) * h] The approximation's accuracy increases as the number of subintervals (n) increases, resulting in a narrower width (h) and a more accurate estimation of the definite integral.

Step by step solution

01

Defining a definite integral and its geometric interpretation

A definite integral is the signed area under a curve (the function f(x)) over a specified interval [a, b]. Geometrically, it can be interpreted as the sum of the areas of the rectangles or other shapes that are enclosed by the function and the x-axis.
02

Introduction to the Trapezoid Rule

The Trapezoid Rule is a numerical integration technique used to approximate definite integrals. Instead of using rectangles as in the Riemann sum, the trapezoid rule uses trapezoids to approximate the underlying area. The area of each trapezoid can be calculated by taking the average of the function values at the two endpoints and multiplying by the width of the interval.
03

Dividing the interval into subintervals

To apply the trapezoid rule, the interval [a, b] is divided into n equal subintervals, each with width h. This width is determined by: h = (b - a) / n
04

Calculating the area of each trapezoid

Next, the area of each individual trapezoid is calculated. The area of each trapezoid can be found using the formula: Area = (1/2) * (f(x_i) + f(x_(i + 1))) * h where x_i and x_(i + 1) are the left and right endpoints of the i-th subinterval, and f(x_i) and f(x_(i + 1)) are the corresponding function values.
05

Approximating the definite integral using the Trapezoid Rule

Finally, add together the areas of all the trapezoids to approximate the definite integral: Approximate integral = 鈭慱{i = 0}^{n - 1} [(1/2)(f(x_i) + f(x_(i + 1))) * h] This sum gives the approximation of the definite integral using the trapezoid rule. The accuracy of the approximation depends on the number of subintervals (n) used. As the number of subintervals increases, the width (h) decreases, and the Trapezoid Rule provides a more accurate approximation of the definite integral.

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