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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 8.8.21
Chapter 8, Problem 8.8.21

19-22. {Use of Tech} Trapezoid Rule approximations. Find the indicated Trapezoid Rule approximations to the following integrals.
21. ∫(0 to 1) sin(πx) dx using n = 6 subintervals

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Understand the Trapezoid Rule formula: The Trapezoid Rule approximates the integral of a function f(x) over [a, b] using n subintervals. The formula is: T = (b - a)/(2n) * [f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(b)], where x₁, x₂, ..., xₙ₋₁ are the points dividing the interval into n subintervals.
Identify the given values: Here, the integral is ∫(0 to 1) sin(πx) dx, the interval is [0, 1], and the number of subintervals is n = 6.
Calculate the width of each subinterval (h): The width is given by h = (b - a)/n. Substituting a = 0, b = 1, and n = 6, compute h.
Determine the x-values for the subintervals: Divide the interval [0, 1] into 6 subintervals using the width h. The x-values will be x₀ = 0, x₁ = h, x₂ = 2h, ..., x₆ = 1. Write down these values explicitly.
Apply the Trapezoid Rule formula: Evaluate f(x) = sin(πx) at each x-value (x₀, x₁, ..., x₆). Substitute these values into the Trapezoid Rule formula to approximate the integral.

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Key Concepts

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

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles, providing a better approximation. The formula involves calculating the average of the function values at the endpoints of each subinterval and multiplying by the width of the subintervals.
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Subintervals

Subintervals are segments into which the interval of integration is divided when applying numerical methods like the Trapezoidal Rule. In this case, with n = 6, the interval from 0 to 1 is divided into six equal parts, each with a width of 1/6. The choice of the number of subintervals affects the accuracy of the approximation.
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Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫(a to b) f(x) dx, where 'a' and 'b' are the limits of integration. The definite integral can be interpreted both geometrically and analytically, and it is fundamental in calculating total quantities such as area, volume, and accumulated change.
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