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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 8.7.4e
Chapter 8, Problem 8.7.4e

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
b. Evaluate the integral directly and find |ET|.
∫ from -2 to 0 of (x² - 1) dx

Verified step by step guidance
1
First, write down the integral to be evaluated directly: \(\int_{-2}^{0} (x^{2} - 1) \, dx\).
Next, find the antiderivative of the integrand \(x^{2} - 1\). Recall that the antiderivative of \(x^{2}\) is \(\frac{x^{3}}{3}\) and the antiderivative of \(-1\) is \(-x\). So, the antiderivative is \(F(x) = \frac{x^{3}}{3} - x\).
Evaluate the antiderivative at the upper and lower limits of integration: calculate \(F(0)\) and \(F(-2)\).
Subtract the values to find the exact value of the integral: \(\int_{-2}^{0} (x^{2} - 1) \, dx = F(0) - F(-2)\).
To find the error bound \(|E_{T}|\) for the Trapezoidal Rule, recall the error formula: \(|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(a = -2\), \(b = 0\), and \(n\) is the number of subintervals used. Compute \(f''(x)\), find its maximum absolute value on \([-2,0]\), and substitute all values into the formula.

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Key Concepts

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

Definite Integral

A definite integral calculates the exact area under a curve between two limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the interval bounds. Evaluating it directly involves finding the antiderivative and applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the area under the curve into trapezoids. It estimates the integral by summing the areas of these trapezoids, which is useful when the exact integral is difficult to compute.
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Error Bound for the Trapezoidal Rule (|ET|)

The error bound |ET| measures the difference between the exact integral and the Trapezoidal Rule approximation. It depends on the second derivative of the function and the number of subintervals, providing a way to estimate the accuracy of the numerical approximation.
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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

b. Evaluate the integral directly and find |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 1 to 3 of (2x - 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

b. Evaluate the integral directly and find |ET|.

∫ from 0 to π of sin(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 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

b. Evaluate the integral directly and find |ET|.

∫ from 1 to 3 of (2x - 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

b. Evaluate the integral directly and find |ET|.

∫ from 1 to 2 of 1 / s² ds