Chapter 5 | Integration
521
Figure 5.14 The graph of f ( x ) =10− x 2 is set up for a right-endpoint approximation of the area bounded by the curve and the x -axis on [1, 2], and it shows a lower sum.
The Riemann sum is ∑ k =1 4 ⎛
⎠ (0.25) = 0.25 ⎡
⎣ 10−(1.25) 2 +10−(1.5) 2 + 10 − (1.75) 2 +10−(2) 2 ⎤ ⎦
⎝ 10− x 2 ⎞
= 0.25[8.4375 + 7.75 + 6.9375 + 6] =7.28.
The area of 7.28 is a lower sum and an underestimate.
5.5
a. Find an upper sum for f ( x ) =10− x 2 on [1, 2]; let n =4. b. Sketch the approximation.
Example 5.6 Finding Lower and Upper Sums for f ( x ) =sin x Find a lower sum for f ( x ) = sin x over the interval ⎡ ⎣ a , b ⎤ ⎦ = ⎡ ⎣ 0, π 2 ⎤
⎦ ; let n =6.
Solution Let’s first look at the graph in Figure 5.15 to get a better idea of the area of interest.
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