# 15.1 - 15.1 DOUBLE INTEGRALS OVER RECTANGLES KIAM HEONG KWA...

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15.1: DOUBLE INTEGRALS OVER RECTANGLES KIAM HEONG KWA 1. Volumes and Double Integrals Let f be a nonnegative continuous function of two variables deﬁned on a closed rectangle R = [ a,b ] × [ c,d ] = { ( x,y ) R 2 | a x b,c y d } , and let S = { ( x,y,z ) R 3 | 0 z f ( x,y ) , ( x,y ) R } be the solid that lies above R and under the graph of f . Before giving a precise meaning to the volume V = V ( S ) of S , we ﬁrst do an approx- imate calculation of how much space the solid S occupies. Of course, a very very very crude approximation can be done by simply choosing at random a sample point ( x * ,y * ) in R and then evaluating the product of f ( x * ,y * ) and the area of R : V f ( x * ,y * ) × the area of R. However, this is usually a very very very bad approximation, unless f varies very little over the rectangle R . Intuitively, it is clear that the variation (in values) of a continuous function decreases as we decrease the size of the region over which the variation is measured. Hence we shall ﬁrst divide the solid S into solids S ij of much smaller dimensions and approximate the volume V ij of each S ij . The precise deﬁnition for S ij will be given in the sequel. We divide the base of

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15.1 - 15.1 DOUBLE INTEGRALS OVER RECTANGLES KIAM HEONG KWA...

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