(a) Left Riemann sum with six subintervals:
Δt = (120 - 0)/6 = 20
Left endpoints: t = 0, 20, 40, 60, 80, 100
f(0) = 1.2, f(20) = 2.8, f(40) = 4.0, f(60) = 4.7, f(80) = 5.1, f(100) = 5.2
Sum = 20 * (1.2 + 2.8 + 4.0 + 4.7 + 5.1 + 5.2) = 20 * 23.0 = 460
(b) Right Riemann sum with six subintervals:
Δt = 20
Right endpoints: t = 20, 40, 60, 80, 100, 120
f(20) = 2.8, f(40) = 4.0, f(60) = 4.7, f(80) = 5.1, f(100) = 5.2, f(120) = 4.8
Sum = 20 * (2.8 + 4.0 + 4.7 + 5.1 + 5.2 + 4.8) = 20 * 26.6 = 532
(c) Midpoint sum with three subintervals:
Δt = (120 - 0)/3 = 40
Midpoints: t = 20, 60, 100
f(20) = 2.8, f(60) = 4.7, f(100) = 5.2
Sum = 40 * (2.8 + 4.7 + 5.2) = 40 * 12.7 = 508
(d) Trapezoidal rule with three subintervals:
Δt = 40
Endpoints: t = 0, 40, 80, 120
f(0) = 1.2, f(40) = 4.0, f(80) = 5.1, f(120) = 4.8
Sum = (40/2) * [f(0) + 2*f(40) + 2*f(80) + f(120)] = 20 * [1.2 + 2*4.0 + 2*5.1 + 4.8] = 20 * [1.2 + 8.0 + 10.2 + 4.8] = 20 * 24.2 = 484
Parent Tip: Review the logic above to help your child master the concept of riemann sum worksheet.
