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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 8.7.4d
Chapter 8, Problem 8.7.4d

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

Verified step by step guidance
1
Identify the integral to approximate: \(\int_{-2}^{0} (x^{2} - 1) \, dx\) and note that we will use the Trapezoidal Rule with \(n = 4\) subintervals.
Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a = -2\) and \(b = 0\).
Determine the \(x\)-values at the endpoints of each subinterval: \(x_0 = -2\), \(x_1 = -2 + \Delta x\), \(x_2 = -2 + 2\Delta x\), \(x_3 = -2 + 3\Delta x\), and \(x_4 = 0\).
Evaluate the function \(f(x) = x^{2} - 1\) at each of these points to get \(f(x_0), f(x_1), f(x_2), f(x_3), f(x_4)\).
Apply the Trapezoidal Rule formula: \(T_n = \frac{\Delta x}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)\right]\) to estimate the integral.
To find an upper bound for the error \(|E_T|\), use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\).
Compute the second derivative \(f''(x)\) of the function \(f(x) = x^{2} - 1\), then find its maximum absolute value on the interval \([-2, 0]\).
Substitute the values of \(a\), \(b\), \(n\), and \(\max |f''(x)|\) into the error bound formula to get the upper bound for \(|E_T|\).

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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 to approximate definite integrals by dividing the interval into n subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas provides an estimate of the integral, improving accuracy as n increases.
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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 gauge the accuracy of the approximation.
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Definite Integral of a Polynomial Function

A definite integral of a polynomial function, like ∫(x² - 1) dx, represents the net area between the curve and the x-axis over a specified interval. Understanding how to integrate polynomials exactly helps in comparing numerical approximations to the true value.
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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 2 of x 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 3 of (2x - 1) dx

Textbook Question

Finding area

Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:

c. 2π ≤ x ≤ 3π.

1
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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
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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 1 to 2 of 1 / s² ds