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Hence

Example 3 Evaluate the lower Riemann sum L ( P,f ) and the upper Riemann sum U ( P,f ) where P = { 0, 1/4, 1/3, 1/2, 3/4, 4/5, 1 } and f ( x ) = arctan x ,

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  1. Example 3 Evaluate the lower Riemann sum L(P,f )and the upper Riemann sum U(P,f )where P = {0, 1/4, 1/3, 1/2, 3/4, 4/5, 1}and f(x) = arctan x, Solution To locate the minimum of f on the subintervals of [0,1] and compute L(P,f ), sketch the graph of f(x)= arctan x. Then t1*=0, the point where f has its minimum value on the subinterval [0,1/4], t2*=1/4, the point where f has its minimum value on [1/4,1/3], t3*=1/3, the point where f has its minimum value on the subinterval [1/3,1/2], t4*=1/2, the point where f has its minimum value on the subinterval [1/2,3/4], t5*=3/4, the point where f has its minimum value on the subinterval [3/4,4/5] and t6*=4/5, the point where f has its minimum value on the subinterval [4/5,1].

  2. t1*=0, t2*=1/4, t3*=1/3, t4*=1/2, t5*=3/4, t6*=4/5 P = {0, 1/4, 1/3, 1/2, 3/4, 4/5, 1} • Hence • L(P,f) = R(P,T*,f) = f(t1*)(x1-x0) + f(t2*)(x2-x1) + f(t3*)(x3-x2) + f(t4*)(x4-x3) + f(t5*)(x5-x4) + f(t6*)(x6-x5) • where x0=0, x1=1/4, x2=1/3, x3=1/2, x4=3/4, x5=4/5, x6=1: • L(P,f) = (arctan 0)(1/4 – 0) + (arctan 1/4)(1/3– ¼)+ (arctan 1/3)(1/2-1/3) + (arctan 1/2)(3/4-1/2) + (arctan 3/4)(4/5-3/4) + (arctan 4/5)(1-4/5) • (0)(.25) + (.245)(.083) + (.322)(.167) + (.464)(.250) + (.644)(.050) + (.675)(.200) • 0+ .020+ .054+ .116 + .032 +.135  0.357

  3. To locate the maximum of f on the subintervals of [0,1] and compute U(P,f ), sketch the graph of f(x)=arctan x. Then t1**=1/4, the point where f has its maximum value on the subinterval [0,1/4], t2**=1/3, the point where f has its maximum value on [1/4,1/3], t3**=1/2, the point where f has its maximum value on the subinterval [1/3,1/2], t4**=3/4, the point where f has its maximum value on the subinterval [1/2,3/4], t5**=4/5, the point where f has its maximum value on the subinterval [3/4,4/5] and t6**=1, the point where f has its maximum value on the subinterval [4/5,1].

  4. t1**=1/4, t2**=1/3, t3**=1/2, t4**=3/4, t5**=4/5, t6**=1 x0=0, x1=1/4, x2=1/3, x3=1/2, x4=3/4, x5=4/5, x6=1 • Hence • U(P,f) = R(P,T**,f) = f(t1**)(x1-x0) + f(t2**)(x2-x1) + f(t3**)(x3-x2) + f(t4**)(x4-x3) + f(t5**)(x5-x4) + f(t6**)(x6-x5). • = (arctan 1/4)(1/4 – 0) + (arctan 1/3)(1/3– ¼) + (arctan 1/2)(1/2-1/3) + (arctan 3/4)(3/4-1/2) + (arctan 4/5)(4/5-3/4) + (arctan 1)(1-4/5) • (.245)(.250) + (.322)(.083) + (.464)(.167) + (.644)(.250) + (.675)(.050) + (.785)(.200) • .061+ .027+ .076+ .161 + .034 +.157  0.516 Let A be the area of the region bounded by the graph of f(x) = arctan x, the x-axis and the line x=1. By Prop. 3.2.32, A  ½ [L(P,f)+U(P,f)] = ½ [ 0.357 + 0.516 ] = 0.437 with error less than ½[U(P,f)-L(P,f)] = ½ [ 0.516 – 0.357 ] = 0.080. Using the Riemann sum of Example 3.2C (3), the inequality L(P,f) R(P,T,f) U(P,f) is illustrated by: 0.357  0.437  0.516

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