Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces the concept of integration as the reverse process of differentiation. Students learn to compute antiderivatives using rules like the power rule, and to handle constants and sums effectively. The chapter emphasizes the use of integration in diverse applications—from physics problems involving displacement and acceleration to economic models of cost and revenue—and introduces integration by substitution as a method to simplify complex integrals.

Learning Objectives

1

Define and identify antiderivatives and indefinite integrals, including proper notation.

2

Apply the power rule for integration and understand its restrictions (n ? -1).

3

Utilize constant multiple and sum/difference rules to integrate various functions.

4

Solve real-world problems using integration, such as determining displacement from velocity and calculating cost or revenue functions.

5

Apply the substitution technique to simplify and solve more complex integrals.

Key Concepts

CONCEPT

DEFINITION

Antiderivative

A function F(x) such that F'(x) = f(x). It represents the reverse process of differentiation.

Indefinite Integral

The general antiderivative of a function, written as ∫ f(x) dx, and expressed as F(x) + C, where C is the constant of integration.

Constant of Integration

An arbitrary constant C that represents the fact that antiderivatives differ by a constant.

Power Rule for Integration

A rule used to compute ∫ x^n dx = x^(n+1)/(n+1) + C for any real number n ≠ -1.

Substitution Method

A technique for integrating composite functions by letting u = g(x) so that du = g'(x) dx, helping to simplify the integrand.

Example Problems

Example 1

What must be true of $F(x)$ and $G(x)$ if both are antiderivatives of $f(x) ?$

Example 2

How is the antiderivative of a function related to the function?

Example 3

Explain what is wrong with the following use of the power rule: $\int \frac{5}{x^{2}} d x=\frac{5}{x^{3} / 3}+C$

Example 4

Explain why the restriction $n \neq-1$ is necessary in the rule $\int x^{n} d x=\frac{x^{n+1}}{n+1}+C$

Example 5

Find the following. Problem 5 - 42 5. $\int 8 d x$
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Step-by-Step Explanations

QUESTION

Find an antiderivative of f(x) = 5x^4.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the power rule for differentiation tells us d/dx(x^n) = n*x^(n-1). To reverse this, increase the exponent by 1 and divide by the new exponent.
Step 2: Since the derivative of x^5 is 5x^4, conclude that an antiderivative is F(x) = x^5.
Step 3: Add the constant of integration, giving F(x) = x^5 + C.
Final Answer: ∫ 5x^4 dx = x^5 + C.

Finding an Antiderivative using the Power Rule

QUESTION

Evaluate ∫ (1/2x^3 + 12·46x^2) dx using the substitution method for a composite function example if necessary.

STEP-BY-STEP ANSWER:

Step 1: Identify a part of the integrand that can be substituted. For instance, if you have an expression like 12x^3 + 1, set u = 2x^3 + 1.
Step 2: Compute du; for example, if u = 2x^3 + 1 then du/dx = 6x^2 so that du = 6x^2 dx.
Step 3: Rewrite the original integral in terms of u and du. This often simplifies the integration to a basic form such as ∫ u^n du.
Step 4: Integrate with respect to u using the power rule and substitute back in terms of x.
Final Answer: The substituted antiderivative will be expressed back in x, complete with the constant of integration C.

Integration by Substitution

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Common Mistakes

  • Forgetting the constant of integration (C) when writing the final antiderivative.
  • Misapplying the power rule for integration when n = -1, which should instead be handled using the natural logarithm.
  • Incorrectly substituting or failing to replace the differential dx appropriately in the substitution method.
  • Overcomplicating a problem that can be solved by applying the constant multiple or sum/difference rules directly.