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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 8.7.9e
Chapter 8, Problem 8.7.9e

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

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1
First, recall the exact value of the integral \( \int_0^{\pi} \sin(t) \, dt \). To evaluate it directly, find the antiderivative of \( \sin(t) \), which is \( -\cos(t) \).
Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits: calculate \( -\cos(\pi) - (-\cos(0)) \).
Simplify the expression to find the exact value of the integral.
Next, to find the error bound \( |E_T| \) for the Trapezoidal Rule, use the error formula for the Trapezoidal Rule: \[ |E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)| \], where \( a = 0 \), \( b = \pi \), and \( n \) is the number of subintervals.
Calculate the second derivative of \( f(t) = \sin(t) \), which is \( f''(t) = -\sin(t) \), then find its maximum absolute value on \( [0, \pi] \). Substitute all values into the error bound formula to express \( |E_T| \).

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Key Concepts

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

Definite Integral and Exact Evaluation

A definite integral calculates the exact area under a curve between two limits. For ∫₀^π sin(t) dt, the integral can be evaluated using the antiderivative of sin(t), which is -cos(t). Substituting the limits gives the exact value, serving as a benchmark for numerical approximations.
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Definition of the Definite Integral

Trapezoidal Rule for Numerical Integration

The Trapezoidal Rule approximates the integral by dividing the interval into subintervals and approximating the area under the curve as trapezoids. It uses linear interpolation between points, providing a simple numerical estimate that improves with more subintervals.
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Additional Rules for Indefinite Integrals

Error Bound for the Trapezoidal Rule (|ET|)

The error bound |ET| estimates the maximum difference between the exact integral and the Trapezoidal Rule approximation. It depends on the second derivative of the function and the number of subintervals, helping assess the accuracy of the numerical method.
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Intro to the Chain Rule Example 1
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

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

∫ from -2 to 0 of (x² - 1) dx

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.

e. u = tan^(-1) ((x - 1)/2)

What is the value of the integral?

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