Chapter 5 | Integration
533
Figure 5.17 For a function that is partly negative, the Riemann sum is the area of the rectangles above the x -axis less the area of the rectangles below the x -axis.
Taking the limit as n →∞, the Riemann sum approaches the area between the curve above the x -axis and the x -axis, less the area between the curve below the x -axis and the x -axis, as shown in Figure 5.18 . Then, ∫ 0 2 f ( x ) dx = lim n →∞ ∑ i =1 n f ( c i )Δ x = A 1 − A 2 . The quantity A 1 − A 2 is called the net signed area .
Figure 5.18 In the limit, the definite integral equals area A 1 less area A 2 , or the net signed area.
Notice that net signed area can be positive, negative, or zero. If the area above the x -axis is larger, the net signed area is positive. If the area below the x -axis is larger, the net signed area is negative. If the areas above and below the x -axis are equal, the net signed area is zero. Example 5.9 Finding the Net Signed Area
Made with FlippingBook - professional solution for displaying marketing and sales documents online