multipl_integrls

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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CV. Changing Variables in Multiple Integrals 1. Changing variables. Double integrals in x, y coordinates which are taken over circular regions, or have inte- grands involving the combination x2 + y2, are often better done in polar coordinates: This involves introducing the new variables r and 19, together with the equations relating them to x, y in both the forward and backward directions: Changing the integral to polar coordinates then requires three steps: A. Changing the integrand f (x, y) to g(r, 8), by using (2); B. Supplying the area element in the r, I9 system: dA = r dr dB ; C. Using the region R to determine the limits of integration in the r, I9 system. In the same way, double integrals involving other types of regions or integrands can sometimes be simplified by changing the coordinate system from x, y to one better adapted to the region or integrand. Let's call the new coordinates u and v; then there will be equations introducing the new coordinates, going in both directions: (often one will only get or use the equations in one of these directions). To change the integral to u,v-coordinates, we then have to carry out the three steps A, B, C above. A first step is to picture the new coordinate system; for this we use the same idea as for polar coordinates, namely, we consider the grid formed by the level curves of the new coordinate functions: I \ ,u=Uo (4) u(x, y) = UO, v(x,y) = vo . Once we have this, algebraic and geometric intuition will usually handle steps A and C, but for B we will need a formula: it uses a determinant called the Jacobian, whose notation and definition are v=v2 Using it, the formula for the area element in the u, v-system is
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2 18.02 NOTES so the change of variable formula is where g(u, v) is obtained from f (x, y) by substitution, using the equations (3). We will derive the formula (5) for the new area element in the next section; for now let's check that it works for polar coordinates. Example 1. Verify (1) using the general formulas (5) and (6) Solution. Using (2), we calculate: so that dA = r dr dB, according to (5) and (6); note that we can omit the absolute value, since by convention, in integration problems we always assume r 2 0, as is implied already by the equations (2). We now work an example illustrating why the general formula is needed and how it is used; it illustrates step C also - putting in the new limits of integration. Example 2. Evaluate JJ, (. dx dy over the region R pictured. 2) Solution. This would be a painful : integral : to work out in rectangular coordinates.
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multipl_integrls - MIT OpenCourseWare http:/ocw.mit.edu...

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