Definite Integral Calculator

How to Use This Calculator

1. Enter f(x), lower bound a, and upper bound b.
2. Set the number of subintervals n (must be even for Simpson's rule).
3. Switch to the Trapezoid tab to compare with the trapezoid rule.
4. Use Compare n to see how accuracy improves with more subintervals.
5. The Professional tab adds adaptive quadrature and error bounds.

Formula

Example

Frequently Asked Questions

• What is the difference between Simpson's rule and the trapezoid rule? ▼
  The trapezoid rule approximates the area using trapezoids (linear interpolation). Simpson's rule uses parabolas (quadratic interpolation) and is typically 4th-order accurate vs 2nd-order for trapezoid.
• How accurate is Simpson's rule with n=1000? ▼
  For a smooth function, Simpson's 1/3 rule with n=1000 gives accuracy of about O(h⁴) = O(10⁻¹²), well beyond the precision needed for most calculations.
• What is the average value of a function on an interval? ▼
  The mean value theorem for integrals states the average value is (1/(b−a))∫ₐᵇ f(x)dx. This represents the height of a rectangle with the same area as under the curve.
• What is signed versus unsigned area? ▼
  The definite integral gives signed area — regions below the x-axis contribute negatively. Unsigned area uses |f(x)| so all regions contribute positively.
• Why does increasing n improve accuracy? ▼
  More subintervals means each parabolic/trapezoidal piece covers a smaller interval, reducing approximation error. The 'Compare n' tab shows how the result converges as n grows.

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