Skip to main content
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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

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.
Recommended video:
Guided course
5:50
Power Rules

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.
Recommended video:
06:11
Introduction to Riemann Sums

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.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

1. Give some examples of analytical methods for evaluating integrals.

119
views
Textbook Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

38. ∫ x² ln²(x) dx

91
views
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

19. ∫[0 to π/3] sin⁵x cos⁻²x dx

68
views
Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

8. ∫ sin 3x cos 2x dx

49
views
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

50. ∫ csc¹⁰x cot³x dx

79
views
Textbook Question

Clever substitution Evaluate ∫ dx/(1 + sin x + cos x) using the substitution x=2 tan⁻¹ θ. The identities sin x = 2 sin(x/2) cos(x/2) and cos x =cos²(x/2) − sin²(x/2) are helpful.

39
views