chapter 15 multiple integrals - abbymath.com · multiple integrals 15-2 & 15-3 iterated...
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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
*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.
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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
1
x y4
x 2 y
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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.
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