chapter 15 multiple integrals - abbymath.com · multiple integrals 15-2 & 15-3 iterated...

Chapter 15 Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates 15-6 Surface Area (4th ed.) 15-6 Triple Integrals 15-7 Triple Integrals in Cylindrical Coordinates 15-8 Triple Integrals in Spherical Coordinates The following notes are for the Calculus D (SDSU Math 252) classes I teach at Torrey Pines High School. I wrote and modified these notes over several semesters. The explanations are my own; however, I borrowed several examples and diagrams from the textbooks* my classes used while I taught the course. Over time, I have changed some examples and have forgotten which ones came from which sources. Also, I have chosen to keep the notes in my own handwriting rather than type to maintain their informality and to avoid the tedious task of typing so many formulas, equations, and diagrams. These notes are free for use by my current and former students. If other calculus students and teachers find these notes useful, I would be happy to know that my work was helpful. - Abby Brown SDUHSD Abby Brown Calculus III/D SDSU Math 252 www.abbymath.com San Diego, CA * , 6th & 4th editions, James Stewart, ©2007 & 1999 Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2. (Chapter, section, page, and formula numbers refer to the 6th edition of this text.) , 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards, Calculus: Early Transcendentals *Calculus ©1994 D. C. Heath and Company, ISBN 0-669-35335-3. Name: ___________________________________ www.abbymath.com - Ch. 15 Page 1 of 15

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Chapter 15Multiple Integrals

15-2 & 15-3 Iterated Integrals15-1, 15-2, & 15-3 Double Integrals and Volume15-4 Double Integrals in Polar Coordinates15-6 Surface Area (4th ed.)15-6 Triple Integrals15-7 Triple Integrals in Cylindrical Coordinates15-8 Triple Integrals in Spherical Coordinates

The following notes are for the Calculus D (SDSU Math 252)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown

SDUHSD

Abby Brown

Calculus III/DSDSU Math 252

www.abbymath.comSan Diego, CA

* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)

, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,

Calculus: Early Transcendentals

Name: ___________________________________ www.abbymath.com - Ch. 15

Page 1 of 15

Page 2: Chapter 15 Multiple Integrals - abbymath.com · Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates

Name: ___________________________________ www.abbymath.com - Ch. 15

Page 2 of 15

Page 3: Chapter 15 Multiple Integrals - abbymath.com · Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates

Name: ___________________________________ www.abbymath.com - Ch. 15

Page 3 of 15

Page 4: Chapter 15 Multiple Integrals - abbymath.com · Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates

How do we decide dxdy or dydx?Consider both (1) the shape of the region and (2) the integrand.Usually one order of integration is easier than the other. To switch the order of integration, sketch the region determined by the limits and use the graph to help you write new limits.

1 2 3

x y4

x 2 y

Name: ___________________________________ www.abbymath.com - Ch. 15

Page 4 of 15

Page 5: Chapter 15 Multiple Integrals - abbymath.com · Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates

Name: ___________________________________ www.abbymath.com - Ch. 15

Page 5 of 15

Page 6: Chapter 15 Multiple Integrals - abbymath.com · Multiple Integrals 15-2 & 15-3 Iterated Integrals 15-1, 15-2, & 15-3 Double Integrals and Volume 15-4 Double Integrals in Polar Coordinates

Given a surface f(x,y), we can approximate the volume under the surface over a given region R in the xy-plane. Break up the region into squares, calculate the height at a corresponding point in each square, calculate the volume of each rectangular prism, and add all of the volumes together. Then change the approximation to an infinite number of prisms by taking the limit.

Name: ___________________________________ www.abbymath.com - Ch. 15

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