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A Riemann sum approximates a definite integral by dividing the area under a curve into rectangles (or trapezoids) and summing their areas. As the number of subdivisions n increases, the approximation converges to the exact integral.
Four common methods: Left Riemann Sum (rectangle height = function value at left endpoint), Right Riemann Sum (height = right endpoint), Midpoint Rule (height = midpoint), and Trapezoidal Rule (trapezoids instead of rectangles — generally more accurate).
Formula for n equal subintervals over [a,b]: Δx = (b−a)/n. Left sum: Σᵢ₌₀ⁿ⁻¹ f(xᵢ)·Δx. Right sum: Σᵢ₌₁ⁿ f(xᵢ)·Δx. Midpoint: Σᵢ₌₁ⁿ f((xᵢ₋₁+xᵢ)/2)·Δx. Trapezoidal: Δx/2 · [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)].
Worked example: Approximate ∫[1→3] x² dx with n=4 using the left sum. Δx = 0.5. Left endpoints: 1, 1.5, 2, 2.5. Sum = 0.5·(1 + 2.25 + 4 + 6.25) = 0.5 · 13.5 = 6.75. Exact value = [x³/3]₁³ = 9 − 1/3 = 8.667. Error = 1.917.
Left: Σ f(xᵢ)·Δx | Trap: (Δx/2)[f(x₀) + 2Σf(xᵢ) + f(xₙ)]
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