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.
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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.
The midpoint rule often provides better accuracy than the left or right Riemann sums for smooth functions.
The error in the midpoint rule approximation decreases as the number of subintervals increases, assuming a continuous second derivative.
It can be particularly useful when dealing with integrals that are difficult or impossible to compute analytically.
The midpoint rule can be applied to both definite and improper integrals, though caution is needed for handling singularities.
Review Questions
How does the midpoint rule differ from left and right Riemann sums?
What is the formula for applying the midpoint rule to approximate an integral?
In what scenarios might you prefer using the midpoint rule over other numerical integration techniques?
A numerical integration technique that approximates the integral by averaging function values at consecutive points and multiplying by the interval width.