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Chapter 5 | Integration
Figure 5.4 In the right-endpoint approximation of area under a curve, the height of each rectangle is determined by the function value at the right of each subinterval. Note that the right-endpoint approximation differs from the left-endpoint approximation in Figure 5.3 .
The graphs in Figure 5.5 represent the curve f ( x ) = x 2 2 . In graph (a) we divide the region represented by the interval [0, 3] into six subintervals, each of width 0.5. Thus, Δ x =0.5. We then form six rectangles by drawing vertical lines perpendicular to x i −1 , the left endpoint of each subinterval. We determine the height of each rectangle by calculating f ( x i −1 ) for i =1,2,3,4,5,6. The intervals are ⎡ ⎣ 0, 0.5 ⎤ ⎦ , ⎡ ⎣ 0.5, 1 ⎤ ⎦ , ⎡ ⎣ 1, 1.5 ⎤ ⎦ , ⎡ ⎣ 1.5, 2 ⎤ ⎦ , ⎡ ⎣ 2, 2.5 ⎤ ⎦ , ⎡ ⎣ 2.5, 3 ⎤ ⎦ . We find the area of each rectangle by multiplying the height by the width. Then, the sum of the rectangular areas approximates the area between f ( x ) and the x -axis. When the left endpoints are used to calculate height, we have a left-endpoint approximation. Thus,
A ≈ L 6 = ∑ i =1 6
f ( x i −1 )Δ x = f ( x 0 )Δ x + f ( x 1 )Δ x + f ( x 2 )Δ x + f ( x 3 )Δ x + f ( x 4 )Δ x + f ( x 5 )Δ x = f (0)0.5+ f (0.5)0.5+ f (1)0.5+ f (1.5)0.5+ f (2)0.5+ f (2.5)0.5 = (0)0.5 + (0.125)0.5 + (0.5)0.5 + (1.125)0.5 + (2)0.5 + (3.125)0.5 = 0 + 0.0625 + 0.25 + 0.5625 + 1 + 1.5625 =3.4375.
Figure 5.5 Methods of approximating the area under a curve by using (a) the left endpoints and (b) the right endpoints.
In Figure 5.5 (b), we draw vertical lines perpendicular to x i such that x i is the right endpoint of each subinterval, and calculate f ( x i ) for i =1,2,3,4,5,6. We multiply each f ( x i ) byΔ x to find the rectangular areas, and then add them. This is a right-endpoint approximation of the area under f ( x ). Thus,
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