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15.1 - Section 16.1 Double Integrals over Rectangles...

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Section 16.1 Double Integrals over Rectangles “Integrating Functions of more than one Variable” In single variable calculus, we developed the definite integral as a way to measure the area under a graph. We shall generalize this idea to multivariable calculus. We shall start by reviewing the single variable definite integral. 1. Review Suppose f ( x ) is a function which is continuous on the interval [ a,b ]. Then to define the definite integral of f over [ a,b ], we do the following: ( i ) Break up the interval into n equally sized pieces of size Δ x = ( b - a ) /n . ( ii ) In each interval, fix some x value in that interval (perhaps the midpoint, the left hand endpoint or the right hand end- point). Observe that the value f ( x i x is a rectangle which approximates the area under f in the i th subinterval. ( iii ) Define the Riemann sum over the interval [ a,b ] as the sum of the areas of these rectangles: n summationdisplay i =1 f ( x i x ( iv ) If we choose more and more subintervals, the value gets closer and closer to the actual area. Thus we define: Definition 1.1. The definite integral of f from a to b is defined to be integraldisplay b a f ( x ) dx = lim n →∞ n summationdisplay i =1 f ( x i x. 2. Volumes and Double Integrals We shall generalize the ideas of the definite integral to functions of two variables. Suppose the region R is a rectangle in the xy -plane: R = [ a,b ] × [ c,d ] = { ( x,y ) R 2 | a lessorequalslant x lessorequalslant b,c lessorequalslant y lessorequalslant d } . We temporarily assume that f ( x,y ) > 0 on this intevral. We want to define a sum which approximates the volume under f ( x,y ). We do this as follows:
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