2: Multiple Integration

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Dec 20, 2024
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- 1.9: Chapter 1 Review Exercises
- 2.1: Double Integrals over Rectangular Regions

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In this chapter we extend the concept of a definite integral of a single variable to double and triple integrals of functions of two and three variables, respectively. We examine applications involving integration to compute volumes, masses, and centroids of more general regions. We will also see how the use of other coordinate systems (such as polar, cylindrical, and spherical coordinates) makes it simpler to compute multiple integrals over some types of regions and functions. In the preceding chapter, we discussed differential calculus with multiple independent variables. Now we examine integral calculus in multiple dimensions. Just as a partial derivative allows us to differentiate a function with respect to one variable while holding the other variables constant, we will see that an iterated integral allows us to integrate a function with respect to one variable while holding the other variables constant.

The City of Arts and Sciences in Valencia, Spain. Details in caption. — Figure $\PageIndex{1}$ : The City of Arts and Sciences in Valencia, Spain, has a unique structure along an axis of just two kilometers that was formerly the bed of the River Turia. The l’Hemisfèric has an IMAX cinema with three systems of modern digital projections onto a concave screen of 900 square meters. An oval roof over 100 meters long has been made to look like a huge human eye that comes alive and opens up to the world as the “Eye of Wisdom.” ( CC-BY-2.0 modification of work by Javier Yaya Tur, Wikimedia Commons)

In the preceding chapter, we discussed differential calculus with multiple independent variables. Now we examine integral calculus in multiple dimensions. Just as a partial derivative allows us to differentiate a function with respect to one variable while holding the other variables constant, we will see that an iterated integral allows us to integrate a function with respect to one variable while holding the other variables constant.

2.1: Double Integrals over Rectangular Regions
In this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xyxy-plane. Many of the properties of double integrals are similar to those we have already discussed for single integrals.
- 2.1E: Exercises
2.2: Double Integrals over General Regions
In this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in this situation as well, but some techniques need to be extended to cover this more general case.
- 2.2E: Exercises
2.3: Double Integrals in Polar Coordinates
Double integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept of a double integral in a polar rectangular region.
- 2.3E: Exercises
2.4: Triple Integrals
In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- 2.4E: Exercises
2.5: Cylindrical and Spherical Coordinates
In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.
- 2.5E: Exercises
2.6: Triple Integrals in Cylindrical and Spherical Coordinates
This page covers the evaluation of triple integrals using cylindrical and spherical coordinates, emphasizing their application in symmetric regions. It explains conversions between coordinates, details the integration process, and showcases examples involving geometric shapes like cylinders, spheres, and paraboloids. The importance of Fubini's Theorem is highlighted for changing integration order. The text also includes exercises for practice and discusses calculating volumes of solid regions.
- 2.6E: Exercises
2.7: Calculating Centers of Mass and Moments of Inertia
In this section we develop computational techniques for finding the center of mass and moments of inertia of several types of physical objects, using double integrals for a lamina (flat plate) and triple integrals for a three-dimensional object with variable density. The density is usually considered to be a constant number when the lamina or the object is homogeneous; that is, the object has uniform density.
- 2.7E: Exercises
2.8: Change of Variables in Multiple Integrals
When solving integration problems, we make appropriate substitutions to obtain an integral that becomes much simpler than the original integral. We also used this idea when we transformed double integrals in rectangular coordinates to polar coordinates and transformed triple integrals in rectangular coordinates to cylindrical or spherical coordinates to make the computations simpler.
- 2.8E: Exercises
2.9: Chapter 2 Review Exercises
This page presents exercises in multivariable calculus covering integral evaluation, area and volume determination, and center of mass calculations. Problems include confirmations of Fubini's theorem, double and triple integrals, and practical applications like Earth modeling and ski resort work estimation. Key answers for selected tasks illustrate results associated with integrals and geometric figures.

Thumbnail: Double integral as volume under a surface $z = 10 − x^2 − y^2/8$ . The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated. (Public Domain; Oleg Alexandrov).

Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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