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 8.7.1d
Chapter 8, Problem 8.7.1d

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from 1 to 2 of x dx

Verified step by step guidance
1
Identify the integral to approximate: \(\int_{1}^{2} x \, dx\) and note that the interval is from \(a=1\) to \(b=2\).
Determine the step size \(h\) using the formula \(h = \frac{b - a}{n}\), where \(n=4\). So, calculate \(h = \frac{2 - 1}{4}\).
Calculate the \(x\)-values at which the function will be evaluated: \(x_0 = 1\), \(x_1 = 1 + h\), \(x_2 = 1 + 2h\), \(x_3 = 1 + 3h\), and \(x_4 = 2\).
Apply the Trapezoidal Rule formula to estimate the integral: \(T_n = \frac{h}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)\right]\), where \(f(x) = x\) in this problem.
To find an upper bound for the error \(|E_T|\), use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\). Calculate \(f''(x)\) for \(f(x) = x\), find its maximum absolute value on \([1,2]\), and substitute all values to get the error bound.

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 to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoid areas provides an estimate of the integral, improving accuracy with more subintervals (n).
Recommended video:
Guided course
5:50
Power Rules

Error Bound for the Trapezoidal Rule

The error bound |ET| for the Trapezoidal Rule estimates the maximum possible difference between the true integral and its approximation. It depends on the second derivative of the function, the interval length, and the number of subintervals, providing a way to assess the accuracy of the approximation.
Recommended video:
04:57
Determining Error and Relative Error

Definite Integral of a Function

A definite integral calculates the exact area under a curve between two points on the x-axis. Understanding the integral of the function f(x) = x from 1 to 2 helps verify numerical approximations and provides a benchmark for evaluating the accuracy of methods like the Trapezoidal Rule.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 1 to 3 of (2x - 1) dx

Textbook Question

89. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / x², y = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.

1
views
Textbook Question

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 0 to π of sin(t) dth

Textbook Question

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 0 to 2 of (t³ + t) dt

Textbook Question

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from -2 to 0 of (x² - 1) dx

Textbook Question

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 1 to 2 of 1 / s² ds