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Chapter 5 | Integration
Figure 5.7 The graph shows the right-endpoint approximation of the area under f ( x ) = x 2 from 0 to 2.
The left-endpoint approximation is 1.75; the right-endpoint approximation is 3.75.
5.4
Sketch left-endpoint and right-endpoint approximations for f ( x ) = 1 x on [1, 2]; use n =4. Approximate the area using both methods.
Looking at Figure 5.5 and the graphs in Example 5.4 , we can see that when we use a small number of intervals, neither the left-endpoint approximation nor the right-endpoint approximation is a particularly accurate estimate of the area under the curve. However, it seems logical that if we increase the number of points in our partition, our estimate of A will improve. We will have more rectangles, but each rectangle will be thinner, so we will be able to fit the rectangles to the curve more precisely. We can demonstrate the improved approximation obtained through smaller intervals with an example. Let’s explore the idea of increasing n , first in a left-endpoint approximation with four rectangles, then eight rectangles, and finally 32 rectangles. Then, let’s do the same thing in a right-endpoint approximation, using the same sets of intervals, of the same curved region. Figure 5.8 shows the area of the region under the curve f ( x ) = ( x −1) 3 +4 on the interval [0, 2] using a left-endpoint approximation where n =4. The width of each rectangle is Δ x = 2−0 4 = 1 2 . The area is approximated by the summed areas of the rectangles, or L 4 = f (0)(0.5)+ f (0.5)(0.5) + f (1)(0.5)+ f (1.5)0.5 =7.5.
Figure 5.8 With a left-endpoint approximation and dividing the region from a to b into four equal intervals, the area under the curve is approximately equal to the sum of the areas of the rectangles.
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