since we have partitions, our Δx will be the ; largest length of our partitions as the following:
π/6 - 0 = π/6 ===> largest
π/4 - π/6 = π/12
π/3 - π/4 = π/12
π/2 - π/3 = π/6 ===> largest
Δx = π/6
Right hand sum:
π/2
∫ xsin(x)dx ≈ [ f(π/6) Δx + f(π/4) Δx + f(π/3) Δx + f(π/2) Δx ]
0
Δx [ f(π/6) + f(π/4) + f(π/3) + f(π/2) ]
(π/6) [ (π/6) sin(π/6) + (π/4) sin(π/4) + (π/3) sin(π/3) + (π/2) sin(π/2) ]
(π/6) [ (π/6) (1/2) + (π/4) (√(2)/2) + (π/3) (√(3)/2) + (π/2) 1 ]
(π/6) ( (π/12) + (π√(2)/8) + (π√(3)/6) + (π/2) ) ≈ 1.73
Left hand sum:
Δx [ f(0) + f(π/6) + f(π/4) + f(π/3) ]
(π/6) [ 0 + (π/6) sin(π/6) + (π/4) sin(π/4) + (π/3) sin(π/3) ]
(π/6) [ (π/6) (1/2) + (π/4) (√(2)/2) + (π/3) (√(3)/2) ]
(π/6) ( (π/12) + (π√(2)/8) + (π√(3)/6) ) ≈ 0.903
Actual :
π/2
∫ xsin(x)dx = 1
0
