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Chapter 5 | Integration
Figure 5.11 Now we divide the area under the curve into four equal subintervals for a right-endpoint approximation.
Dividing the region over the interval [0, 2] into eight rectangles results in Δ x = 2−0 8 =0.25.
The graph is shown in
Figure 5.12 . The area is
R 8 = 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) + f (2)(0.25) =8.25.
Figure 5.12 Here we use right-endpoint approximation for a region divided into eight equal subintervals.
Last, the right-endpoint approximation with n =32 is close to the actual area ( Figure 5.13 ). The area is approximately R 32 = f (0.0625)(0.0625) + f (0.125)(0.0625) + f (0.1875)(0.0625) + ⋯ + f (2)(0.0625) =8.0625.
Figure 5.13 The region is divided into 32 equal subintervals for a right-endpoint approximation.
Based on these figures and calculations, it appears we are on the right track; the rectangles appear to approximate the area under the curve better as n gets larger. Furthermore, as n increases, both the left-endpoint and right-endpoint approximations appear to approach an area of 8 square units. Table 5.1 shows a numerical comparison of the left- and right-endpoint
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