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Unformatted text preview: Unit 5 Area Between Curves We have learned how to find the area beneath a positive valued func tion, by evaluating a definite integral. In this section we will learn how to find more general areas, of regions lying between two curves. For example, suppose we wish to find the area of the region bounded by f ( x ) and g ( x ) between x = 1 and x = 3 as shown: f(x) g(x) 1 2 3 x y dx f(x)  g(x) Figure 1: Area bounded by two curves The height of the region whose area we want to find is given by f ( x ) g ( x ). And on the interval [1,3], we have f ( x ) ≥ g ( x ), so the function ( f ( x ) g ( x )) is nonnegative on [1,3]. With this observation in mind, we can see that when f ( x ) ≥ g ( x ) the problem of finding the area between two curves f ( x ) and g ( x ) between x = a and x = b can be reduced to one of finding the 1 area under the nonnegative function h ( x ) = f ( x ) g ( x ) between x = a and x = b . Therefore we do not need to learn another technique for this kind of problem, we simply reduce the problem as above, and then apply our earlier method for finding the area under a nonnegative function on an interval [ a, b ] by evaluating the definite integral of that function from a to b . Example 1 . Find the area between the curves f ( x ) = x 2 +1 and g ( x ) = x 1 on the interval [1 , 3]. Solution: We can first sketch the region involved, as shown in figure 1 on the previous page. We see that f ( x ) ≥ g ( x ) throughout [1,3], so the function h ( x ) = f ( x ) g ( x ) is positive valued throughout [1,3]. We may therefore represent the area ( A ) by: A = integraltext 3 1 [ f ( x ) g ( x )] dx = integraltext 3 1 [( x 2 + 1) ( x 1)] dx = integraltext 3 1 [( x 2 x + 2)] dx = bracketleftBig x 3 3 x 2 2 + 2 x bracketrightBig 3 1 = [(9 9 2 + 6) ( 1 3 1 2 + 2)] = 21 2 11 6 = 26 3 Whenever we need to find the area between 2 curves, our first step must always be to determine which curve lies above the other on the interval we are interested in. That is, we need to identify the upper curve and the lower curve, in order to set up the integral integraltext b a ( upper lower ) dx . We must also be sure that the same curve is the upper curve throughout the entire interval. (We’ll look at what to do when this is not the case a bit later.) Often we are interested in the area of the region “bounded by” two curves. If 2 curves, y = f ( x ) and y = g ( x ), intersect in exactly 2 places, at x = a and 2 x = b , and both curves are continuous on [ a, b ], then there is a single region which is entirely enclosed by these curves (see, for instance, figure 2). This is what we mean by the area “bounded by” the curves. If the curves intersect more than twice, then there will be more than one such region. In that case, the area of the region bounded by the curves means the total area of all regions enclosed by the curves....
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 Fall '09
 OLDS,VICKY
 Calculus, Graph of a function

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