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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 8.7.5e
Chapter 8, Problem 8.7.5e

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

Verified step by step guidance
1
First, write down the integral to be evaluated: \(\int_0^2 (t^3 + t) \, dt\).
To evaluate the integral directly, find the antiderivative of the integrand. Recall that the antiderivative of \(t^3\) is \(\frac{t^4}{4}\) and the antiderivative of \(t\) is \(\frac{t^2}{2}\).
Express the antiderivative as \(F(t) = \frac{t^4}{4} + \frac{t^2}{2}\).
Apply the Fundamental Theorem of Calculus by evaluating \(F(t)\) at the upper limit and subtracting the value at the lower limit: \(F(2) - F(0)\).
To find the error bound \(|E_T|\) for the Trapezoidal Rule, use the formula \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq t \leq b} |f''(t)|\), where \(f(t) = t^3 + t\). Compute the second derivative \(f''(t)\), find its maximum on \([0, 2]\), 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.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the area under a curve into trapezoids rather than rectangles. It estimates the integral by averaging the function values at the endpoints of subintervals and multiplying by the subinterval width, providing a better approximation than simple Riemann sums.
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Exact Evaluation of Definite Integrals

Exact evaluation involves finding the precise value of a definite integral using antiderivatives and the Fundamental Theorem of Calculus. For the integral of t³ + t from 0 to 2, one finds the antiderivative, evaluates it at the bounds, and subtracts to get the exact area under the curve.
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Definition of the Definite Integral

Error Bound for the Trapezoidal Rule (|ET|)

The error bound |ET| for the Trapezoidal Rule estimates the maximum difference between the approximate and exact integral values. It depends on the second derivative of the function and the width of the subintervals, providing a way to assess 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 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

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

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

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 x dx