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Unformatted text preview: Section 16.3 Double Integrals over General Regions Not every region is a rectangle In the last two sections we considered the problem of integrating a function of two variables over a rectangle. This situation however is fairly unrealistic since most regions in 2-space are not rectangles. We need to develop a way to calculate integrals over general regions in the plane. In order to integrate a function f ( x, y ) over an arbitrary region R , we shall develop two related methods dependent upon the structure of the region R . 1. Integrals of types 1 and 2 The first type of region we consider is when the boundary of the region consists of two functions of x . We describe below. ( i ) Suppose we want to integrate a function f ( x, y ) over some arbitrary region R in the plane which is bounded between two functions g 1 ( x ) and g 2 ( x ) and between a and b as illustrated below. We call such a region a region of type 1 . a g1 g2 b ( ii ) We shall set up a Riemann sum over this region to evaluate the definite integral. First we subdivide the region up into small rectangles as illustrated where we do not count the rectangles which are not fully contained in R : b g1 g2 a ( iii ) As usual, we take reference points in each subrectangle, and take a Riemann sum over all the rectangles multiplying the area of the rectangle by the function value at the reference 1 2 point in the rectangle. Notice however, that this case is dif- ferent to the case when we are integrating over rectangles - specifically, the number of rectangles and the values y is al- lowed to take depend upon x . Specifically, if we choose x i as a reference point along a column of rectangles, then the y values are bounded between g 1 ( x i ) and g 2 ( x i ), so all rectan- gles must be within these bounds. This must be reflected in any Riemann sum we are using to try to define an integral. Therefore, we can define integraldisplay integraldisplay R f ( x, y ) dA n summationdisplay i =1 m summationdisplay j =1 f ( x i , y j ) y x where the second sum is over the rectangles bounded by g 1 ( x i ) and g 2 ( x i ). ( iv ) Taking smaller and smaller rectangles, the answer becomes more exact because the rectangles approximate the region bet- ter and better. Taking them infinitely small, we get the fol- lowing iterated integrals: integraldisplay integraldisplay R f ( x, y ) dA = integraldisplay b a integraldisplay g 2 ( x ) g 1 ( x ) f ( x, y ) dydx....
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