Approximate integral of 0 to pi/2 with function xsin(x) by computing Lf(P) (left

Question

Approximate integral of 0 to pi/2 with function xsin(x) by computing Lf(P) (left side) and Uf(P) (right side), using the partition {0,pi/6,pi/4,pi/3,pi/2 }. (The answer as a formula is ok)

(last problem had question marks...)

Explanation / Answer

since we have partitions, our ?x will be the ; largest length of our partitions as the following: p/6 - 0 = p/6 ===> largest p/4 - p/6 = p/12 p/3 - p/4 = p/12 p/2 - p/3 = p/6 ===> largest ?x = p/6 Right hand sum: p/2 ? xsin(x)dx ˜ [ f(p/6) * ?x + f(p/4) * ?x + f(p/3) * ?x + f(p/2) * ?x ] 0 ?x * [ f(p/6) + f(p/4) + f(p/3) + f(p/2) ] (p/6) * [ (p/6) * sin(p/6) + (p/4) * sin(p/4) + (p/3) * sin(p/3) + (p/2) * sin(p/2) ] (p/6) * [ (p/6) * (1/2) + (p/4) * (v(2)/2) + (p/3) * (v(3)/2) + (p/2) * 1 ] (p/6) * ( (p/12) + (pv(2)/8) + (pv(3)/6) + (p/2) ) ˜ 1.73 Left hand sum: ?x * [ f(0) + f(p/6) + f(p/4) + f(p/3) ] (p/6) * [ 0 + (p/6) * sin(p/6) + (p/4) * sin(p/4) + (p/3) * sin(p/3) ] (p/6) * [ (p/6) * (1/2) + (p/4) * (v(2)/2) + (p/3) * (v(3)/2) ] (p/6) * ( (p/12) + (pv(2)/8) + (pv(3)/6) ) ˜ 0.903 Actual : p/2 ? xsin(x)dx = 1 0 ================== 2) a = 4 b = 8 n = 4 terms ?x = [ b - a ] / n ------>?x = [ 8 - 4 ] / 4 = 4/4 = 1 8 ? ((4/x)+5) dx ˜ [ f(5) * ?x + f(6) * ?x + f(7) * ?x + f(8) * ?x ] 4 ?x * [ f(5) + f(6) + f(7) + f(8) ] 1 * [ ((4/5)+5) + ((4/6)+5) + ((4/7)+5) + ((4/8)+5) ] [ (29/5) + (34/6) + (39/7) + (44/8) ] = (4733/210) ˜ 22.5 Left hand sum: ?x * [ f(4) + f(5) + f(6) + f(7) ] 1 * [ ((4/4)+5) + ((4/5)+5) + ((4/6)+5) + ((4/7)+5) ] [ 6 + (29/5) + (34/6) + (39/7) ] = (2419/105) ˜ 23.04 Actual: 8 ? ((4/x)+5) dx ˜ 22.8 4 Please le me know if there is any question :)

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