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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 17 Double and iterated integrals in the plane March 13 Reading Material: From Simmons : 20.1 and 20.2. From Course Notes : I. Last time: Nonindependent variables. Partial differential equations. Today: Double and iterated integrals in the plane. 2 Integrals of functions of one variable Consider a function y = f ( x ) such that x is in [ a, b ]. Assume also in this case that f ( x ) 0 on [ a, b ]. Then we know that the area A of the sub graph is given by Area A = Z b a f ( x ) dx We also know how to evaluate this integral using an antiderivative F ( x ) ( F ( x ) = f ( x )). In fact by the fundamental theorem of calculus A = Z b a f ( x ) dx = F ( b ) F ( a ) A couple of questions are natural at this point: Question: Why areas are computed through integrals? What does an integral really mean? Take the interval [ a, b ] and subdivide it into smaller intervals by introducing a partition = { a = x , x 1 , ..., x i , ..., x n = b } 1 Call I i = [ x i , x i +1 ] and x i = x i +1 x i . For each interval I i we pick a point x * i in it and we measure the hight h i = f ( x * i ) ....
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This note was uploaded on 03/08/2010 for the course MATH 18.022 taught by Professor Hartleyrogers during the Spring '06 term at MIT.
 Spring '06
 HartleyRogers
 Equations, Integrals, Multivariable Calculus

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