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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 8.7.5d
Chapter 8, Problem 8.7.5d

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

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1
Identify the integral to approximate: \(\int_0^2 (t^3 + t) \, dt\) and note that we will use the Trapezoidal Rule with \(n=4\) subintervals.
Calculate the width of each subinterval using the formula \(\Delta t = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5\).
Determine the partition points: \(t_0 = 0\), \(t_1 = 0.5\), \(t_2 = 1.0\), \(t_3 = 1.5\), and \(t_4 = 2.0\).
Evaluate the function \(f(t) = t^3 + t\) at each partition point: compute \(f(t_0)\), \(f(t_1)\), \(f(t_2)\), \(f(t_3)\), and \(f(t_4)\).
Apply the Trapezoidal Rule formula: \(T_n = \frac{\Delta t}{2} \left[f(t_0) + 2f(t_1) + 2f(t_2) + 2f(t_3) + f(t_4)\right]\) to estimate the integral.
To find the upper bound for the error \(|E_T|\), first find the second derivative of the function: \(f''(t) = \frac{d^2}{dt^2}(t^3 + t)\).
Determine the maximum value of \(|f''(t)|\) on the interval \([0, 2]\).
Use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq t \leq b} |f''(t)|\) to calculate the upper bound.

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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 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 the number of subintervals increases.
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Error Bound for the Trapezoidal Rule

The error bound 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.
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Definite Integral of Polynomial Functions

Definite integrals of polynomial functions, like t³ + t, can be computed exactly using antiderivatives. Understanding the behavior and derivatives of polynomials helps in applying numerical methods and calculating error bounds effectively.
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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

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

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

b. Evaluate the integral directly and find |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

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

∫ from 1 to 2 of 1 / s² ds

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