key term - Midpoint rule

Definition

The midpoint rule is a numerical integration technique used to approximate the definite integral of a function. It estimates the integral by taking the value of the function at the midpoint of each subinterval and multiplying it by the width of the subintervals.

5 Must Know Facts For Your Next Test

1. The formula for the midpoint rule is $M_n = \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right) \Delta x$, where $\Delta x$ is the width of each subinterval.
2. The midpoint rule often provides better accuracy than the left or right Riemann sums for smooth functions.
3. The error in the midpoint rule approximation decreases as the number of subintervals increases, assuming a continuous second derivative.
4. It can be particularly useful when dealing with integrals that are difficult or impossible to compute analytically.
5. The midpoint rule can be applied to both definite and improper integrals, though caution is needed for handling singularities.

Review Questions

Related terms

Trapezoidal Rule:

A numerical integration technique that approximates the integral by averaging function values at consecutive points and multiplying by the interval width.

Simpson's Rule:

A numerical method that approximates an integral by fitting parabolas through segments of data points.

Riemann Sum:

An approximation of an integral obtained by summing up areas of rectangles under a curve. Variants include left, right, and mid-point sums.