121616949-math.191 - g x =-x 3 7 x 2-10 x 5 over the...

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9 Applications of Integration We have seen how integration can be used to find an area between a curve and the x -axis. With very little change we can find some areas between curves; indeed, the area between a curve and the x -axis may be interpreted as the area between the curve and a second “curve” with equation y = 0. In the simplest of cases, the idea is quite easy to understand. EXAMPLE 9.1 Find the area below f ( x ) = - x 2 + 4 x + 3 and above
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Unformatted text preview: g ( x ) =-x 3 + 7 x 2-10 x + 5 over the interval 1 ≤ x ≤ 2. In figure 9.1 we show the two curves together, with the desired area shaded, then f alone with the area under f shaded, and then g alone with the area under g shaded. It is clear from the figure that the area we want is the area under f minus the area under g , which is to say Z 2 1 f ( x ) dx-Z 2 1 g ( x ) dx = Z 2 1 f ( x )-g ( x ) dx. 177...
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