Integral Calc

How Integral Calc Works

∫ₐᵇ f(x)dx ≈ (h/3)[f(a) + f(b) + 4 Σ f(x_odd) + 2 Σ f(x_even)], h = (b−a)/n, n even

f(x)

Integrand — The function to be integrated, expressed in terms of the variable x.

a

Lower limit — The starting point of the integration interval.

b

Upper limit — The ending point of the integration interval.

n

Subdivisions — The number of subintervals used for the numerical approximation. Must be even for Simpson's rule.

h

Step size — The width of each subinterval, computed as h = (b − a) / n.

How to Use

1. Enter the function f(x) using x as the variable. Use ^ for powers (e.g., x^2 for x squared), * for explicit multiplication, and parentheses for grouping (e.g., (x^2 + 1) / (x - 1)).
2. Enter the lower bound a and upper bound b to define the integration interval. The calculator automatically sorts a and b so lower ≤ upper.
3. Optionally adjust the number of subdivisions n (default 100). Use higher values (e.g., 500–1000) for greater accuracy, especially for oscillatory functions.
4. The result shows the approximate integral using Simpson's rule, along with the trapezoidal rule result for comparison. A small discrepancy between the two methods indicates high confidence in the result.
5. The step-by-step work panel breaks down the computation: step size h, endpoint values, sums of odd/even terms, and the final Simpson formula application.

Quick Reference

Common Uses

• •Computing areas under curves in physics (e.g., work from force–distance graph, displacement from velocity)
• •Calculating probabilities and cumulative distribution functions in statistics
• •Determining total accumulated quantities in engineering (charge, energy, fluid volume)
• •Verifying analytic integration done by hand in calculus coursework
• •Computing arc lengths, volumes of revolution, and centroids in geometry

Understanding the Result

Frequently Asked Questions

Why must the number of subdivisions be even?

Simpson's 1/3 rule fits a quadratic polynomial through every three consecutive sample points (two adjacent subintervals). This requires an even total number of subintervals. If you enter an odd value, the calculator automatically increments it by 1 to make it even.

How does Simpson's rule compare to the trapezoidal rule?

Simpson's rule is significantly more accurate for smooth functions. The trapezoidal rule uses straight-line segments (linear interpolation) between points, giving error O(h²). Simpson's rule uses parabolic arcs (quadratic interpolation), giving error O(h⁴). For most smooth functions, Simpson's rule achieves the same accuracy with far fewer subdivisions.

What does the method discrepancy tell me?

The discrepancy between Simpson and trapezoidal results gives a rough error estimate. If the two methods agree closely (discrepancy < 0.001), you can have high confidence. A large discrepancy suggests the function has high curvature or irregularities — try increasing n or breaking the interval at points of rapid change.

What happens if my function has a singularity inside the interval?

Numerical integration methods including Simpson's rule assume the function is finite and well-behaved throughout [a, b]. If your function has a division by zero, vertical asymptote, or discontinuity within the interval, the result will be inaccurate or NaN. Consider splitting the interval at the singularity or using an improper integral technique.

Can I integrate exponential or trigonometric functions?

Yes — the basic expression parser supports standard arithmetic operators (+, -, *, /, ^) and parentheses. For example, you can integrate e^x by entering e^x if your expression uses Euler's number... wait, the basic parser does not support the constant "e" or "sin"/"cos". For those, see the Limit Calculator which supports trigonometric and exponential functions.

Sources

• Wikipedia — Simpson's Rule
• Wolfram MathWorld — Numerical Integration