Answer:
9.29
Step-by-step explanation:
S₄ is the area using Simpson's rule and 4 intervals.
Simpson's rule can be calculated as:
Sᵢ = Δx/3 (f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 4f(xᵢ₋₂) + 2f(xᵢ₋₁) + f(xᵢ))
Or, Simpson's rule can be calculated as a combination of midpoint and trapezoid sums:
S₂ᵢ = (2Mᵢ + Tᵢ) / 3
Using the first method:
Δx = (4−0)/4 = 1
S₄ = 1/3 (1.00 + 4(1.41) + 2(2.24) + 4(3.16) + 4.12)
S₄ = 1/3 (27.88)
S₄ ≈ 9.29
Using the second method:
S₄ = (2M₂ + T₂) / 3
The midpoint area for 2 intervals is:
M₂ = (2) (1.41) + (2) (3.16) = 9.14
The trapezoid area for 2 intervals is:
T₂ = ½ (1.00 + 2.24) (2) + ½ (2.24 + 4.12) (2) = 9.60
Therefore:
S₄ = (2 (9.14) + 9.60) / 3
S₄ ≈ 9.29
