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Unformatted text preview: Chapter 14: Multiple Integrals Summary: The previous chapter focused upon taking derivatives and limits of multivariable functions. In this chapter, the integration of multivariable functions is considered. This naturally leads to integrating functions with respect to two or three variables. The integration of these types of functions over their domains is described best using double and triple integrals. Double integrals are useful for calculating the area of a two dimensional region or its center of mass. They are also useful for calculating the surface area of a surface determined by a function z = f ( x , y ) . Triple integrals allow the mass of an object to be determined and consequently the location of its center of mass. In many problems, situations in two and three dimensions are not easily described using Cartesian or rectangular coordinates. In some cases, problems may be more easily considered if the coordinates are given as polar coordinates in two dimen- sions or by cylindrical coordinates or spherical coordinates in three dimensions. Just as u-substitution played an important role in the development of integration in one dimension, using a change of variables is important in both double and triple integral problems. In particular, when switching from rectangular coordinates to either polar, cylindrical or spherical coordinates, a change of variables is being performed to switch from one set of coordinates to another. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Set up double integrals and evaluate double integrals (§14 . 1). 2. Write double integrals with nonconstant limits of integration (§14 . 2). 3. Set up double integrals over nonrectangular regions (using rectangular, §14 . 2, or polar coordinates, §14 . 3). 4. Find the volume under a surface using double integrals (§14 . 1 − 14 . 3). 5. Switch the order of integration in double integrals (§14 . 2 , 14 . 3). 6. Change between rectangular and polar coordinates in double integrals (§14 . 3). 7. Calculate area using double integrals (§14 . 2 , 14 . 3). 8. Define surfaces parametrically (§14 . 4). 307 308 9. Express a parametric surface in vector form (§14 . 4). 10. Find tangent planes to parametric surfaces and surface area of a parametric surface (§14 . 4). 11. Calculate volume under a surface using a triple integral (§14 . 5). 12. Determine appropriate limits of integration for triple integrals (§14 . 5). 13. Set up and evaluate triple integrals using both cylindrical and spherical co- ordinates (§14 . 6). 14. Convert between different coordinate systems for triple integrals (§14 . 6). 15. Find the Jacobian and perform general changes of variables in double and triple integrals (§14 . 7)....
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This note was uploaded on 06/01/2010 for the course CAL 1000 taught by Professor Lee during the Spring '10 term at École Normale Supérieure.
- Spring '10