Rectangular Rule Integration
Approximates a definite integral over [a, b] by dividing the interval into equal rectangles and summing their areas. Ideal for numerical integration when an exact antiderivative is difficult or impossible to find.
About this calculator
The rectangular (midpoint or left/right endpoint) rule approximates the definite integral ∫ₐᵇ f(x) dx by partitioning the interval [a, b] into n equal subintervals of width Δx = (b − a) / n. Each subinterval is treated as a rectangle whose height is a representative function value f(x*). The total area — and hence the integral approximation — is: ∫ ≈ Δx × f(x*) × n = (b − a) / n × functionValue × n, which simplifies here to: result = (upperBound − lowerBound) / intervals × functionValue. As n increases, the rectangles become thinner and the approximation converges to the true integral. This method is foundational in numerical analysis and underpins more advanced techniques like the trapezoidal and Simpson's rules.
How to use
Suppose you want to approximate ∫₀⁴ f(x) dx with an average function value of 3 over 8 intervals. Enter lowerBound = 0, upperBound = 4, intervals = 8, and functionValue = 3. The calculator computes: (4 − 0) / 8 × 3 = 0.5 × 3 = 1.5 per rectangle. Multiplied across all 8 intervals the total approximation is 1.5 × 8 = 12. Increasing n improves accuracy — try 16 intervals to see the estimate stabilize.
Frequently asked questions
How accurate is the rectangular rule for numerical integration?
The rectangular rule is a first-order method, meaning its error decreases proportionally to the width Δx as you add more intervals. For smooth functions, doubling the number of intervals roughly halves the approximation error. It is less accurate than the trapezoidal rule or Simpson's rule for the same number of intervals, but it is simple to implement and understand. For highly irregular or discontinuous functions, accuracy depends heavily on where within each subinterval the function value is sampled.
What is the difference between the left, right, and midpoint rectangular rules?
These three variants differ only in which x-value within each subinterval is used as the rectangle's height. The left rule uses the left endpoint, the right rule uses the right endpoint, and the midpoint rule uses the center of each subinterval. The midpoint rule generally produces the smallest error of the three because it captures the function's average behavior more accurately. All three converge to the true integral as the number of intervals n approaches infinity.
When should I use numerical integration instead of an exact antiderivative?
Numerical integration is necessary when a function has no closed-form antiderivative — for example, e^(−x²) or sin(x)/x. It is also useful when f(x) is defined only by tabulated data points rather than an explicit formula. Engineers, physicists, and data scientists rely on numerical methods like the rectangular rule whenever analytical integration is impractical. Choosing a sufficiently large number of intervals ensures the approximation meets the required precision for the application.