Chapter 5 | Integration
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Figure5.9 shows the same curve divided into eight subintervals. Comparing the graph with four rectangles in Figure5.8 with this graph with eight rectangles, we can see there appears to be less white space under the curve when n =8. This white space is area under the curve we are unable to include using our approximation. The area of the rectangles is
L 8 = f (0)(0.25) + f (0.25)(0.25) + f (0.5)(0.25) + f (0.75)(0.25) + f (1)(0.25) + f (1.25)(0.25) + f (1.5)(0.25) + f (1.75)(0.25) =7.75.
Figure 5.9 The region under the curve is divided into n =8 rectangular areas of equal width for a left-endpoint approximation.
The graph in Figure 5.10 shows the same function with 32 rectangles inscribed under the curve. There appears to be little white space left. The area occupied by the rectangles is L 32 = f (0)(0.0625) + f (0.0625)(0.0625) + f (0.125)(0.0625) + ⋯ + f (1.9375)(0.0625) =7.9375.
Figure 5.10 Here, 32 rectangles are inscribed under the curve for a left-endpoint approximation.
We can carry out a similar process for the right-endpoint approximation method. A right-endpoint approximation of the same curve, using four rectangles ( Figure 5.11 ), yields an area R 4 = f (0.5)(0.5) + f (1)(0.5)+ f (1.5)(0.5) + f (2)(0.5) =8.5.
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