📐 MathematicsGraph

Riemann Sums

Approximate definite integrals with left, right, and midpoint rectangle rules. See how increasing the number of subdivisions improves accuracy.

Function (0=x², 1=sin x, 2=x³−3x+3, 3=eˣ)

Lower bound a

Upper bound b

Number of rectangles n

Method (0=Left, 1=Right, 2=Midpoint)

• A Riemann sum approximates ∫ f(x)dx as a sum of rectangle areas
• Left rule: uses f at the left edge — overestimates for increasing functions
• Right rule: uses f at the right edge — underestimates for increasing functions
• Midpoint rule: uses f at the midpoint — generally more accurate than left/right
• Error → 0 as n → ∞; the definite integral is the limit of the Riemann sum

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