Riemann Sum Calculator
Approximate definite integrals using left, right, or midpoint Riemann sums. See how accuracy improves as the number of rectangles N increases toward the true integral value.
∫ Calculus📐 ∫ ≈ Δx × Σ f(x*𝑖)🔢 Math
Method
Coefficient c
Exponent n
Lower limit a
Upper limit b
Number of rectangles N
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Riemann Sum Calculator | — | ∫ ≈ Δx × Σ f(x*𝑖) | approximation |
Step-by-Step Examples
Example 1
Left N=4
∫[0,4]x² left N=4
- Δx=1, x=0,1,2,3
- Sum=0+1+4+9=14, exact=21.33, error=34%
✓ 14 (underestimate)
Example 2
More rects
Same integral N=100
- Sum≈20.93, error≈1.9%
- 4x rects → ~10x less error
✓ ≈20.93
Example 3
Midpoint beats left
N=4 midpoint
- x=0.5,1.5,2.5,3.5
- Sum=21, error=1.5% vs 34% for left!
✓ 21 (much better)
Real-World Applications
Numerics
When no closed-form antiderivative exists.
Physics
Work and impulse from discrete data.
Finance
Numerical option pricing.
Data Science
AUC approximation for ROC curves.
Common Mistakes to Avoid
⚠️
Too few rectangles
N=4 gives 30%+ error. Use N≥100 for usable accuracy.
⚠️
Left/right bias
Left underestimates increasing functions; right overestimates.
⚠️
Signed area
Rectangles below x-axis have negative contribution.
Frequently Asked Questions
What is a Riemann sum? ▾
Approximation of ∫[a,b]f using N rectangles of width Δx=(b-a)/N.
Most accurate method? ▾
Midpoint > Trapezoidal > Left/Right for smooth functions.
How does limit give the integral? ▾
As N→∞, Riemann sum converges to exact integral.
Trapezoidal rule? ▾
Average of left and right sum: Δx/2·[f(a)+2f(x₁)+...+f(b)].
Simpson's rule? ▾
Parabolic segments; error O(Δx⁴). Highly accurate.
Related Math Calculators
Formula Explorer connections
Interpretation: This formula measures limiting behavior, instantaneous change, accumulation or multivariable variation. Assumption: The function must satisfy the continuity or differentiability conditions required by the method. Check domains, bounds, orientation and singularities.