121616949-math.193

# 121616949-math.193 - x ≤ 2 these are the same curves as...

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9.1 Area between curves 179 This is exactly the sort of sum that turns into an integral in the limit, namely the integral Z 2 1 f ( x ) - g ( x ) dx. Of course, this is the integral we actually computed above, but we have now arrived at it directly rather than as a modiﬁcation of the diﬀerence between two other integrals. In that example it really doesn’t matter which approach we take, but in some cases this second approach is better. 0 5 10 0 1 2 3 ........................................................................................................................................................................................ . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................................... Figure 9.2 Area between curves. EXAMPLE 9.2 Find the area below f ( x ) = - x 2 + 4 x + 1 and above g ( x ) = - x 3 + 7 x 2 - 10 x + 3 over the interval 1
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Unformatted text preview: x ≤ 2; these are the same curves as before but lowered by 2. In ﬁgure 9.3 we show the two curves together. Note that the lower curve now dips below the x-axis. This makes it somewhat tricky to view the desired area as a big area minus a smaller area, but it is just as easy as before to think of approximating the area by rectangles. The height of a typical rectangle will still be f ( x i )-g ( x i ), even if g ( x i ) is negative. Thus the area is Z 2 1-x 2 + 4 x + 1-(-x 3 + 7 x 2-10 x + 3) dx = Z 2 1 x 3-8 x 2 + 14 x-2 dx. This is of course the same integral as before, because the region between the curves is identical to the former region—it has just been moved down by 2. EXAMPLE 9.3 Find the area between f ( x ) =-x 2 + 4 x and g ( x ) = x 2-6 x + 5 over the interval 0 ≤ x ≤ 1; the curves are shown in ﬁgure 9.4 . Generally we should interpret...
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