Regular partition

Definition

A regular partition of an interval $[a, b]$ is a division of this interval into subintervals of equal length. It is used to approximate areas under curves by dividing the area into rectangles or other simple shapes.

5 Must Know Facts For Your Next Test

1. In a regular partition, the length of each subinterval is given by $\Delta x = \frac{b - a}{n}$, where $n$ is the number of subintervals.
2. Regular partitions are commonly used in Riemann sums to approximate the definite integral.
3. The endpoints of the subintervals in a regular partition are $x_i = a + i \Delta x$ for $i = 0, 1, ..., n$.
4. For larger values of $n$, regular partitions provide more accurate approximations for integrals.
5. A regular partition leads to uniform widths for all rectangles in methods like Left Riemann Sum, Right Riemann Sum, and Midpoint Rule.

Review Questions

• How do you calculate the width of each subinterval in a regular partition?
• Why are regular partitions useful for approximating definite integrals?
• Given an interval $[2, 10]$ and 4 subintervals, what are the endpoints of these subintervals?

Related terms

Riemann Sum: A method for approximating the total area under a curve by summing up areas of multiple rectangles.

$\Delta x$ (Delta x): The width of each subinterval in a regular partition; calculated as $\frac{b - a}{n}$.

Definite Integral: \text{The limit of Riemann sums as } n \rightarrow \infty; represents the exact area under a curve from } [a,b].