Midpoint Rule Calculator – Numerical Integration
Approximate integrals using the midpoint rule method.
How to Use
- Enter the function f(x) using x as the variable
- Specify the lower and upper bounds of integration
- Enter the number of subintervals (n)
- Click calculate to get the approximation
What is the Midpoint Rule?
The midpoint rule is a numerical integration technique used to approximate the definite integral of a function. It works by dividing the interval [a, b] into n equal subintervals and approximating the area under the curve using rectangles whose heights are determined by the function value at the midpoint of each subinterval.
The formula is: ∫f(x)dx ≈ Δx × [f(x₁) + f(x₂) + ... + f(xₙ)], where Δx = (b-a)/n and xᵢ is the midpoint of the i-th subinterval.
Accuracy of the Midpoint Rule
The midpoint rule typically provides better accuracy than the left or right Riemann sums because the midpoint often gives a better representation of the average function value over each subinterval.
- Increasing the number of subintervals improves accuracy
- The error decreases proportionally to 1/n²
- Works well for smooth, continuous functions
- May struggle with functions that have rapid oscillations
Supported Functions
This calculator supports common mathematical functions:
- Basic operations: +, -, *, /, ^ (power)
- Trigonometric: sin(x), cos(x), tan(x)
- Exponential: exp(x), e^x
- Logarithmic: log(x), ln(x)
- Other: sqrt(x), abs(x)
- Constants: pi, e
Frequently Asked Questions
- How many subintervals should I use?
- More subintervals generally give better accuracy. Start with 10-20 for a rough estimate, and increase to 100+ for higher precision. The error decreases as n² increases.
- What functions can I enter?
- Use x as the variable. Supported operations include basic arithmetic (+, -, *, /), powers (^), trigonometric functions (sin, cos, tan), sqrt, abs, log, ln, exp, and constants pi and e.
- Why is my result different from the exact integral?
- The midpoint rule provides an approximation. The difference between the approximation and exact value is called the error. Increase the number of subintervals to reduce this error.
- When should I use the midpoint rule?
- Use it when you need to approximate integrals that are difficult or impossible to solve analytically, or when working with experimental data points.