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Unformatted text preview: LESSON 14 Multiple and Iterated Integrals READ: Section16.1 NOTES: The notion of an integral of a function of one variable is certainly a central topic in calculus. In this section, the idea of integration is extended to functions of several variables. It would be a good idea to quickly review how integrals of a function of a single variable are defined. You have been computing such integrals for such a long time by using the Fundamental Theorem of Calculus, that the original definition of the integral has probably evaporated. See sections 5.1, 5.2 of the text. If you understand the construction in the one variable case, the extension to two or more variables will be easier to follow. The extension of the definition of integration to functions of several variables is most easily visualized for functions of two variables, so we’ll look at a function z = f ( x,y ) defined for points ( x,y ) in a rectangle R = [ a,b ] × [ c,d ] in the x , yplane. We will assume that all the values of f ( x,y ) are bigger than or equal to 0. This assumption is made only to aid our geometric visualization. Everything we do would work for functions of any number of variables with values that are both positive and negative. The integral of f ( x,y ) over the region R is denoted by the symbol ZZ R f ( x,y ) dA. This is called a double integral . The dA is there to remind us that this integral is being computed over an area in the x , yplane. To compute this double integral, the region R is split into little subrectangles by putting dividing lines in parallel to the sides of the rectangle. To keep the algebra easy, we will put the dividing lines in at equally spaced intervals in each direction. Let’s say we put in m 1 vertical dividing lines and n 1 horizontal ones, as in figure 4, page 856 of the text. As you can see, this mesh we thus create splits the rectangle R into mn subrectangles. Next, from each subrectangle, S , select a point ( x * ,y * ). The area of each subrectangle S is multiplied by f ( x * ,y * ), and these terms are all added up. The total is called a Riemann sum for f on R . Geometrically, the product of the area of S and f ( x * ,y * ) represents the volume of a rectangular column built on S and having height f ( x * ,y * ). The sum of the volumes of all these little columns gives an estimate for the volume of the region with base...
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 Winter '11
 JerryMetzger
 Calculus, Integrals, Riemann, Single variable

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