Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces the use of integration to compute areas between curves and volumes of solids. It explains how to construct Riemann sums and take limits to obtain exact areas, emphasizes the importance of identifying the top and bottom functions correctly, and shows how to handle cases where curves intersect. Real-world applications such as analyzing velocity differences between moving vehicles, quantifying income inequality with the Lorenz curve and Gini index, and computing volumes through cross-sectional slicing illustrate the power and versatility of integration.

Learning Objectives

1

Explain how to find the area between two curves using definite integrals and Riemann sums.

2

Set up and evaluate integrals when the top and bottom curves change over an interval.

3

Apply integration techniques to solve real-world problems such as computing distances from velocity curves and measuring income inequality with Lorenz curves and the Gini index.

4

Understand how to approximate volumes of solids by slicing them into cross-sectional areas.

Key Concepts

CONCEPT

DEFINITION

Area Between Curves

The area of a region bounded by two curves y = f(x) and y = g(x) between x = a and x = b is given by the integral ∫[a to b] |f(x) - g(x)| dx. When one function is always above the other, it simplifies to ∫[a to b] [f(x) - g(x)] dx.

Riemann Sum

An approximation of the area under a curve, calculated by summing the areas of rectangles with width Δx and heights determined by function values at chosen sample points.

Cross-Sectional Area

The area A(x) of a slice of a solid taken perpendicular to an axis; used to calculate the volume of a solid by integrating these areas along the axis.

Lorenz Curve

A graphical representation of the distribution of income or wealth within a population, plotting the cumulative percentage of total income received against the cumulative percentage of recipients.

Gini Index

A measure of income inequality defined as twice the area between the Lorenz curve and the line of perfect equality (y = x). A Gini index of 0 indicates perfect equality, while a value closer to 1 indicates high inequality.

Example Problems

Example 1

Find the area of the shaded region.

Example 2

Find the area of the shaded region.

Example 3

Find the area of the shaded region.

Example 4

Find the area of the shaded region.

Example 5

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $ y = e^x $ , $ y = x^2 - 1 $ , $ x = -1 $ , $ x = 1 $
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Step-by-Step Explanations

QUESTION

Find the area of the region bounded above by y = e^x, below by y = x, and vertically by x = 0 and x = 1.

STEP-BY-STEP ANSWER:

Step 1: Identify the top and bottom functions in the interval. Here, y = e^x is above y = x for 0 ≤ x ≤ 1.
Step 2: Set up the integral for the area A: A = ∫[0 to 1] (e^x - x) dx.
Step 3: Compute the integral. The integral of e^x is e^x, and the integral of x is (1/2)x^2.
Step 4: Evaluate the antiderivatives from 0 to 1: A = [e^x - (1/2)x^2] from 0 to 1 = (e - 1/2) - (1 - 0).
Step 5: Simplify the expression: A = e - 1/2 - 1 = e - 3/2.
Final Answer: The area of the region is e - 3/2.

Example 1: Area of Region Bounded by y = e^x and y = x on [0, 1]

QUESTION

When the curves intersect so that one is not always on top, how do you set up the area integral?

STEP-BY-STEP ANSWER:

Step 1: Determine the points of intersection by solving f(x) = g(x) for x.
Step 2: Divide the interval into subintervals where one function is consistently above the other.
Step 3: Set up separate integrals for each subinterval: A = ∫[a to c] |f(x) - g(x)| dx + ∫[c to b] |f(x) - g(x)| dx.
Step 4: Evaluate each integral separately and sum the absolute areas.
Final Answer: The total area is the sum of the integrals over the subintervals, ensuring that the top minus bottom function is consistently taken.

Example 2: Area When Curves Cross

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Common Mistakes

  • Failing to correctly identify which function is on top over the interval, leading to a negative area.
  • Not splitting the integral correctly when the curves intersect and switch positions.
  • Ignoring the absolute value when setting up the integral for areas, which can lead to cancellation of areas.
  • Misapplying the limits of integration or mismatching the variable of integration when switching from x to y.
  • Overlooking the need to use proper approximation methods (like the Midpoint Rule) when exact intersections cannot be found analytically.