Riemann Sums — Approximating Integrals
Left Riemann Sum: Use left endpoints. Underestimates if f is increasing.
Right Riemann Sum: Use right endpoints. Overestimates if f is increasing.
Midpoint: Use midpoints of subintervals. Generally most accurate.
Trapezoidal: Average of left and right. Area of trapezoids.
Formula (equal subintervals):
Δx = (b-a)/n
Left: Σ f(xᵢ)·Δx for i = 0 to n-1
Right: Σ f(xᵢ)·Δx for i = 1 to n
Trap: (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]
Unequal subintervals: The AP exam gives tables with unequal spacing. Multiply each f-value by its specific Δx.
Over/underestimate: Know when each method overestimates or underestimates based on whether f is increasing/decreasing and concave up/down.
Reference:
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