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# 265L15 - LESSON 15 Double Integrals Over General Regions...

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LESSON 15 Double Integrals Over General Regions READ: Section 16.2 NOTES: Integrating a function of two variables over a rectangular region in the x , y -plane is not always sufficient for applications. It might be necessary, for example, to compute an integral of the form RR D f ( x, y ) dA , where D is a circular disk in the x , y -plane. The bad news is that, in general, there is no nice way to compute such double integrals without going back to the definition of the integral as a limit of Riemann sums as described in section 16.1, at least when the region has a very complicated boundary. The good news is that for most types of regions you are likely to meet in real life, there is often an easy way to do the integral which is very similar to the method used to evaluate a double integral over a rectangle. The idea is again to convert the double integral into an iterated integral. One type of region that can be handled is the area trapped between two functions of x , say y = g 2 ( x ) and y = g 1 ( x ) for a x b , as pictured in figure 5, page 869 of the text. Let’s call that region D . Suppose z = f ( x, y ) is the equation for a surface over the region D , and we want to compute RR D f ( x, y ) dA . The reasoning of the last lesson can be repeated. The plan is to compute the volume of the solid under the surface above the region D by first finding a formula for the area of the cross-sections produced when the planes x = c intersects the solid, and then integrate the cross-sections for all the x ’s from a to b . If the solid is sliced by the plane, as described above, through a particular x , the cross-section will be a plane region bounded by the curve z = f ( x, y ) on the top (remember, x is fixed there, and y is the variable in f ( x, y )), with the y values in the interval g 1 ( x ) to g 2 ( x ). That is the key for doing the double integral. In

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