Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.7.13a
Chapter 8, Problem 8.7.13a

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx

Verified step by step guidance
1
Identify the function to be integrated: \(f(x) = x^{2} + 1\) over the interval \([-1, 1]\).
Recall the error bound formula for the Trapezoidal Rule: \(|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.
Compute the second derivative of the function: \(f''(x) = \frac{d^{2}}{dx^{2}}(x^{2} + 1) = 2\).
Determine the maximum value of \(|f''(x)|\) on the interval \([-1, 1]\), which is \(2\) since \(f''(x)\) is constant.
Set up the inequality for the error bound to be less than \(10^{-4}\) and solve for \(n\): \(\frac{(1 - (-1))^{3}}{12 n^{2}} \times 2 < 10^{-4}\). Simplify and solve for \(n\) to find the minimum number of subintervals needed.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8m
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 to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas estimates the integral, with accuracy improving as the number of subintervals increases.
Recommended video:
Guided course
5:50
Power Rules

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule depends on the second derivative of the function being integrated. Specifically, the error magnitude is at most (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps estimate how many subintervals are needed for a desired accuracy.
Recommended video:
04:57
Determining Error and Relative Error

Second Derivative and Its Role in Error Estimation

The second derivative of the integrand measures the function's concavity and affects the error in the Trapezoidal Rule. A larger maximum second derivative on the interval implies a larger potential error, requiring more subintervals to achieve a given accuracy. Calculating or bounding this derivative is essential for applying the error formula.
Recommended video:
06:02
The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.

Textbook Question

81. Find the values of p for which each integral converges.

a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]

Textbook Question

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

a. u = 1/(x + 1)

What is the value of the integral?

Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 2 of 1/s² ds

Textbook Question

Evaluate the integrals in Exercises 33–36.

∫ [1 / (x(9 - x²))] dx

Textbook Question

Evaluate ∫ sec θ dθ by:

a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.