Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter section introduces iterated integrals as a powerful tool for evaluating functions of two variables. Students learn to compute integrals in stages—first integrating with respect to one variable and then the other—while setting up appropriate limits for various types of regions (vertically or horizontally simple). The section emphasizes the importance of visualizing the integration region, switching orders when beneficial, and understanding the link between iterated integrals, area, and volume computations.

Learning Objectives

1

Evaluate iterated integrals by integrating with respect to one variable at a time.

2

Use iterated integrals to compute the area of plane regions, including vertically simple and horizontally simple regions.

3

Determine appropriate limits of integration by sketching the region of interest.

4

Switch the order of integration when necessary and understand its impact on the integration process.

5

Relate the concept of double integrals to finding volumes of solids.

Key Concepts

CONCEPT

DEFINITION

Iterated Integral

An integral computed by performing successive integrations with respect to one variable at a time; typically written without brackets, e.g., ∫[a,b] (∫[g1(x), g2(x)] f(x,y) dy) dx.

Vertically Simple Region

A region in the plane where any vertical line intersects the region in a single interval, allowing the region to be expressed as { (x,y) | x is between a and b and y is between two functions of x }.

Horizontally Simple Region

A region where any horizontal line crosses it in a single interval, so the region can be described with y between constants and x between functions of y.

Double Integral

An integral used to compute the accumulation (such as area or volume) over a two-dimensional region, typically set up as an iterated integral.

Region of Integration

The set of points (x,y) over which an iterated integral is evaluated, determined by the limits of integration.

Example Problems

Example 1

Evaluate the integral. $$\int_{0}^{x}(x+2 y) d y$$

Example 2

Evaluate the integral. $$\int_{x}^{x^{2}} \frac{y}{x} d y$$

Example 3

Evaluate the integral. $$\int_{1}^{2 y} \frac{y}{x} d x, y>0$$

Example 4

Evaluate the integral. $$\int_{0}^{\cos y} y d x$$

Example 5

Evaluate the integral. $$\int_{0}^{\sqrt{4-x^{2}}} x^{2} y d y$$
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Step-by-Step Explanations

QUESTION

Evaluate the iterated integral ∫ from x=1 to 2 ∫ from y=1 to x (2xy) dy dx.

STEP-BY-STEP ANSWER:

Step 1: Identify the inner integral: ∫ from y=1 to x (2xy) dy, treating x as a constant.
Step 2: Factor constant 2x out of the inner integral: 2x ∫ from y=1 to x y dy.
Step 3: Compute the inner integral: ∫ from y=1 to x y dy = (1/2)y^2 evaluated from 1 to x = (1/2)(x^2 - 1).
Step 4: Multiply by 2x: 2x * (1/2)(x^2 - 1) = x(x^2 - 1).
Step 5: Now evaluate the outer integral: ∫ from x=1 to 2 x(x^2 - 1) dx.
Step 6: Expand the integrand: x^3 - x.
Step 7: Compute ∫ from x=1 to 2 (x^3 - x) dx = [ (1/4)x^4 - (1/2)x^2 ] from x=1 to 2.
Step 8: Evaluate at x = 2: (1/4)*16 - (1/2)*4 = 4 - 2 = 2; at x = 1: (1/4)*1 - (1/2)*1 = 0.25 - 0.5 = -0.25.
Step 9: Subtract: 2 - (-0.25) = 2.25.
Final Answer: The value of the iterated integral is 2.25 (or 9/4).

Evaluating an Iterated Integral

QUESTION

Set up an iterated integral to find the area of a vertically simple region R defined by x between a and b and y between g1(x) and g2(x).

STEP-BY-STEP ANSWER:

Step 1: Recognize that the area of a region R can be found by integrating 1 over the region.
Step 2: For a vertically simple region, express the limits for y as functions of x: y goes from g1(x) to g2(x).
Step 3: The outer integration is with respect to x from a to b.
Step 4: Write the iterated integral as: Area = ∫ from x=a to b [∫ from y=g1(x) to g2(x) 1 dy] dx.
Final Answer: The area of R is given by the iterated integral ∫[a,b] (∫[g1(x), g2(x)] dy) dx.

Setting Up an Iterated Integral for Area

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Common Mistakes

  • Using a variable of integration in the limits (e.g., writing a limit that depends on the variable being integrated).
  • Failing to sketch the region of integration, which can lead to incorrect limits.
  • Assuming that switching the order of integration changes the value of the integral; while it may affect the complexity of the evaluation, the final value remains the same if done correctly.
  • Neglecting the role of functions that serve as variable limits when describing vertical or horizontal regions.