Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the fundamental techniques of finding antiderivatives and indefinite integrals, emphasizing the relationship between differentiation and integration. Key methods include applying basic integration rules, rewriting integrals into more manageable forms, and incorporating the constant of integration. Additionally, the section shows how to obtain the general solution to differential equations and narrow it down to a particular solution using initial conditions. Topics such as sigma notation further expand the application of integration in approximating areas and establishing the groundwork for the Fundamental Theorem of Calculus.

Learning Objectives

1

Write the general solution of a differential equation using antiderivatives and the constant of integration.

2

Apply basic integration rules (such as the power rule, constant multiple rule, and sum rule) to find antiderivatives.

3

Determine particular solutions of differential equations using initial conditions.

4

Reformulate integrals into simpler forms (e.g., rewriting or using fractional exponents) to facilitate integration.

5

Interpret sigma notation for sums and understand its role in approximating area under a curve.

Key Concepts

CONCEPT

DEFINITION

Antiderivative

A function F(x) is an antiderivative of f(x) if F'(x) = f(x) for all x in an interval. The family of all such functions is written as F(x) + C, where C is an arbitrary constant.

Indefinite Integral

Another name for the antiderivative of a function, denoted by ∫f(x)dx = F(x) + C.

Constant of Integration

The arbitrary constant C added to an antiderivative, representing the fact that there are infinitely many antiderivatives differing by a constant.

Differential Equation

An equation that involves an unknown function and its derivatives. Its general solution is given in terms of an antiderivative plus a constant.

Initial Condition

A given value of a function at a specific point (usually expressed as F(xâ‚€) = yâ‚€) used to determine the particular solution from the general solution.

Sigma Notation

A concise way to represent sums using the sigma symbol (∑). It is used to write series or sums in a compact form, which is useful in approximating areas and in various applications of integration.

Basic Integration Rules

Rules that provide formulas for integrating common functions, including the power rule, integration of constants, and the sum and constant multiple rules.

Example Problems

Example 1

In Exercises 1 and 2, verify the statement by showing that the derivative of the right side equals the integrand of the left side. $$ \int\left(-\frac{6}{x^{4}}\right) d x=\frac{2}{x^{3}}+C $$

Example 2

In Exercises 1 and 2, verify the statement by showing that the derivative of the right side equals the integrand of the left side. $$ \int\left(8 x^{3}+\frac{1}{2 x^{2}}\right) d x=2 x^{4}-\frac{1}{2 x}+C $$

Example 3

In Exercises 3–6, find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d t}=9 t^{2} $$

Example 4

In Exercises 3–6, find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d t}=5 $$

Example 5

In Exercises 3–6, find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d x}=x^{3 / 2} $$
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Step-by-Step Explanations

QUESTION

Find the general antiderivative of f(x) = 2.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the derivative of 2x is 2.
Step 2: Write the antiderivative as F(x) = 2x + C, where C is the constant of integration.
Final Answer: ∫2 dx = 2x + C.

General Antiderivative of a Function

QUESTION

Evaluate the integral ∫x^3 dx by rewriting it in the proper form.

STEP-BY-STEP ANSWER:

Step 1: Confirm the integrand is already in a form that matches the power rule; x^3 is equivalent to x^3.
Step 2: Apply the power rule: ∫x^n dx = x^(n+1)/(n+1) + C, for n ≠ -1.
Step 3: Substitute n = 3 to get ∫x^3 dx = x^(3+1)/(3+1) + C = x^4/4 + C.
Final Answer: ∫ x^3 dx = x^4/4 + C.

Rewriting Before Integrating

QUESTION

Given the differential equation dy/dx = x^(-2) and the initial condition F(1)=0, find the particular solution.

STEP-BY-STEP ANSWER:

Step 1: Find the general antiderivative: ∫x^(-2) dx = -x^(-1) + C.
Step 2: Use the initial condition F(1)=0: -1 + C = 0.
Step 3: Solve for C: C = 1.
Final Answer: The particular solution is F(x) = -1/x + 1.

Finding a Particular Solution

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Common Mistakes

  • Forgetting to include the constant of integration (C) when writing the general antiderivative.
  • Incorrectly applying the power rule, especially when the exponent is ?1, which requires the natural logarithm function instead of a simple power rule.
  • Integrating terms in the numerator and denominator separately without properly rewriting the expression.
  • Not simplifying the integrand before attempting to integrate, leading to overly complicated or erroneous results.