Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces multiple integrals, extending the concept of integration from areas under curves to volumes under surfaces and beyond. It covers the construction of double integrals via Riemann sums, the use of iterated integrals enabled by Fubini’s Theorem, and encompasses techniques for both rectangular and nonrectangular regions. Additionally, alternative coordinate systems like polar, cylindrical, and spherical coordinates are discussed as tools for simplifying complex integrals. Overall, understanding these concepts is vital for tackling problems in multivariable calculus and their applications in engineering, physics, and related fields.

Learning Objectives

1

Describe and compute double integrals as limits of Riemann sums over rectangular and general regions.

2

Explain how double and triple integrals extend the concept of signed area to signed volume and higher-dimensional applications.

3

Apply Fubini’s Theorem to convert double integrals into iterated integrals in either order of integration.

4

Utilize coordinate transformations such as polar, cylindrical, or spherical coordinates to simplify the evaluation of multiple integrals.

Key Concepts

CONCEPT

DEFINITION

Double Integral

An extension of the definite integral to functions of two variables, representing signed volume, computed as the limit of approximating Riemann sums over a region.

Riemann Sum

A sum obtained by partitioning a region into small subregions, multiplying the function value at chosen sample points by the area of each subregion, and then summing these products.

Iterated Integral

A double integral computed by performing two successive integrations with respect to each variable separately, often justified by Fubini’s Theorem.

Fubini’s Theorem

A theorem that states that if a function is continuous on a rectangular region, its double integral can be computed as an iterated integral in either order of integration.

Triple Integral

An integral extended to functions of three variables, which is often used to compute volumes of solids in three-dimensional space.

Example Problems

Example 1

In Exercises $1-14,$ evaluate the iterated integral. $$ \int_{1}^{2} \int_{0}^{4} 2 x y d y d x $$

Example 2

In Exercises $1-14,$ evaluate the iterated integral. $$ \int_{0}^{2} \int_{-1}^{1}(x-y) d y d x $$

Example 3

In Exercises $1-14,$ evaluate the iterated integral. $$ \int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y $$

Example 4

In Exercises $1-14,$ evaluate the iterated integral. $$ \int_{0}^{1} \int_{0}^{1}\left(1-\frac{x^{2}+y^{2}}{2}\right) d x d y $$

Example 5

In Exercises $1-14,$ evaluate the iterated integral. $$ \int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x $$
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Step-by-Step Explanations

QUESTION

How do you compute the double integral of f(x, y) over a rectangle R defined by a ≤ x ≤ b and c ≤ y ≤ d?

STEP-BY-STEP ANSWER:

Step 1: Partition the rectangle R into smaller subrectangles of area ΔA = Δx Δy.
Step 2: Select a sample point (x_k, y_k) in each subrectangle.
Step 3: Form the Riemann sum by calculating Σ f(x_k, y_k) ΔA.
Step 4: Take the limit as the maximum subrectangle dimensions approach zero to obtain the double integral.
Final Answer: The double integral is defined as ∬_R f(x, y) dA = lim(∥P∥→0) Σ f(x_k, y_k) ΔA.

Double Integral over a Rectangle

QUESTION

How does Fubini’s Theorem help in evaluating a double integral over a rectangular region?

STEP-BY-STEP ANSWER:

Step 1: Verify that the function f(x, y) is continuous on the rectangular region R.
Step 2: Express the double integral as an iterated integral, choosing either order: ∬_R f(x, y) dA = ∫[c to d](∫[a to b] f(x, y) dx) dy or vice versa.
Step 3: Evaluate the inner integral using the Fundamental Theorem of Calculus while treating the other variable as constant.
Step 4: Evaluate the outer integral to obtain the final result.
Final Answer: Fubini’s Theorem allows the double integral to be evaluated as iterated integrals in any order.

Fubini’s Theorem

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Common Mistakes

  • Incorrectly determining the limits of integration, especially in iterated integrals where the limits for one variable may depend on the other.
  • Confusing the order of integration or misassigning the differential elements (dx dy vs dy dx).
  • Overlooking the significance of the area element, particularly when transforming coordinates.
  • Assuming that the Riemann sum converges without verifying the conditions of integrability for the function.