What is the Midpoint Rule in calculus?

The Midpoint Rule is a numerical integration method that approximates the definite integral of a function by dividing the interval into subintervals and using rectangles whose heights are determined by the function value at each subinterval's midpoint. It's generally more accurate than the Left or Right Riemann sums because midpoints often provide better average representation of each subinterval.

How accurate is the Midpoint Rule?

The Midpoint Rule is more accurate than basic Left or Right Riemann sums and comparable to the Trapezoidal Rule for similar numbers of subintervals. Accuracy improves as you increase the number of subintervals (n). The error decreases proportionally to 1/n². For many functions, the Midpoint Rule's approximation is very close to the actual integral value with even a modest number of subintervals.

What's the formula for the Midpoint Rule?

The Midpoint Rule formula is: ∫[a to b] f(x)dx ≈ Δx × [f(x₁*) + f(x₂*) + ... + f(xₙ*)], where Δx = (b-a)/n is the width of each subinterval, and xᵢ* is the midpoint of the i-th subinterval. Each midpoint is calculated as xᵢ* = a + (i - 0.5)Δx for i = 1, 2, ..., n.

When should I use the Midpoint Rule instead of other methods?

Use the Midpoint Rule when you need a simple yet relatively accurate numerical approximation of an integral, especially when the function is smooth and doesn't have extreme variations. It's excellent for educational purposes and quick calculations. For higher accuracy requirements, consider Simpson's Rule or adaptive methods. For functions with known antiderivatives, exact integration is preferred when possible.

How many subintervals should I use?

More subintervals increase accuracy but also increase calculation time. For simple functions, n = 10-50 often provides good approximations for learning purposes. For scientific calculations requiring precision, use n = 100 or more. The optimal number depends on the function's behavior and your accuracy requirements. Test different values to find a balance between accuracy and computational efficiency.

Math

Midpoint Rule Calculator

Approximate the definite integral of a function using the Midpoint Rule.

Function f(x)

Lower Bound (a)

Upper Bound (b)

Subintervals (n)

Enter parameters to calculate.

How to Use This Calculator

Define Your Function and Interval

Enter the function f(x) you want to integrate and specify the integration interval [a, b] - the lower limit (a) and upper limit (b). Make sure your function is defined and continuous (or at least piecewise continuous) over this interval for meaningful results.

Choose Number of Subintervals

Specify n, the number of subintervals to divide your interval into. More subintervals generally mean better accuracy but more computation. Start with n = 10 or 20 for a reasonable approximation, then increase if more precision is needed.

Calculate the Approximation

Click calculate. The calculator will divide [a, b] into n equal subintervals, find the midpoint of each, evaluate f(x) at each midpoint, multiply by the subinterval width, and sum all values to approximate the definite integral.

Interpret Results

Review the approximated integral value. The calculator may also show the calculation steps, including Δx, each midpoint, and function values. Compare with the exact integral if known, or try different n values to see how accuracy improves with more subintervals.