lim_itratd_intgr - MIT OpenCourseWare http:/

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MIT OpenCourseWare 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: .
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I. Limits in Iterated Integrals For most students, the trickiest part of evaluating multiple integrals by iteration is to put in the limits of integration. Fortunately, a fairly uniform procedure is available which works in any coordinate system. You must always begin by sketching the region; in what follows we'll assume you've done this. ' 1. Double integrals in rectangular coordinates. Let's illustrate this procedure on the first case that's usually taken up: double integrals in rectangular coordinates. Suppose we want to evaluate over the region R pictured the integral R = region between x2 + y2 = 1 and x + y = 1 ; we are integrating first with respect to y. Then to put in the limits, 1. Hold x fixed, and let y increase (since we are integrating with respect to y). As the point (x, y) moves, it traces out a vertical line. 2. Integrate from the y-value where this vertical line enters the region R, to the y-value where it leaves R. 3. Then let x increase, integrating from the lowest x-value for which the vertical line intersects R, to the highest such x-value. Carrying out this program for the region R pictured, the vertical line enters R where y = 1 - x, and leaves where y = dm. I A The vertical lines which intersect R are those between x = 0 and x = 1. Thus we get for the limits: JJ, f(x,Y) dydx = To calculate the double integral, integrating in the reverse order f (x, y) dx dy, 1. Hold y fixed, let x increase (since we are integrating first with respect to x).
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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lim_itratd_intgr - MIT OpenCourseWare http:/

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