L13_1 - Spring 2004 Math 253/501503 13 Multiple Integrals...

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Spring 2004 Math 253/501–503 13 Multiple Integrals 13.1 Double Integrals over Rectangles Thu, 19/Feb c ± 2004, Art Belmonte Summary Single Riemann Integral First, a little review from Calc 1. Let f be a function deﬁned on I = [ a , b ]. Split [ a , b ]into m subintervals (typically of equal length) whose endpoints constitute a partition P : a = x 0 < x 1 < x 2 < ··· < x m - 1 < x m = b . Let x * i be a point in the i th subinterval, ± x i - 1 , x i ² .Nowle tthe number of subintervals increase indeﬁnitely, while the maximum size of the subintervals (the norm of P ) shrinks to 0. If the limit lim k P k→ 0 m X i = 1 f ( x * i ) 1 x i exists, we call it the deﬁnite integral of f from a to b , written R b a f ( x ) dx , and say that f is integrable . Z b a f ( x ) = lim k P k→ 0 m X i = 1 f ( x * i ) 1 x i Double Riemann Integral In Calc 3, we similarly deﬁne the double integral of a function f ( x , y ) over a rectangular region R = { ( x , y ) : a x b , c y d } by “slicing and dicing,” as it were. (Think of mincing that onion with your Ginsu knife. ..) That is, split [ a , b m subintervals and [ c , d n subintervals. (Typically, the x -intervals are equal-length, as are the y -subintervals.) The norm k P k of the resulting partition P is the length of the longest diagonal among the subrectangles of the partition. We then form a double Riemann sum and take the limit as k P k shrinks to 0. If this limit exists, we obtain the double integral of f over R . ZZ R f ( x , y ) dA = Z b a Z d c f ( x , y ) dydx = Z d c Z b a f ( x , y ) dxdy = lim k P k→ 0 m X i = 1 n X j = 1 f ³ x * i , y * j ´ 1 x i 1 y j Remarks We’ll actually compute a double integral exactly in this handout by having MATLAB take the limit of a double Riemann sum. Since this is rather difﬁcult to do by hand, we’ll often merely use a particular double Riemann sum as an approximation to a double integral.

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L13_1 - Spring 2004 Math 253/501503 13 Multiple Integrals...

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