Partition

Definition

In mathematical analysis, a partition is a division of an interval into smaller subintervals, which helps in approximating the area under a curve. This concept is crucial for defining the Riemann integral as it establishes how the interval is broken down to calculate Riemann sums, which serve as approximations of the integral. The choice of partition directly affects the accuracy of these approximations and highlights properties of integrable functions.

5 Must Know Facts For Your Next Test

1. A partition divides an interval [a, b] into 'n' subintervals, which are often denoted as [x_{0}, x_{1}], [x_{1}, x_{2}], ..., [x_{n-1}, x_{n}] where a = x_{0} and b = x_{n}.
2. The finer the partition, or the more subintervals you create, the closer the Riemann sums will approximate the true value of the integral.
3. Different types of partitions can be used, such as uniform partitions (where all subintervals are equal) and non-uniform partitions (where subintervals vary in size).
4. The concept of partition is essential for establishing criteria for Riemann integrability, specifically through conditions like boundedness and continuity.
5. Partitions play a key role in showing that if a function is Riemann integrable, then the limit of Riemann sums approaches the same value regardless of how the interval is partitioned.

Review Questions

• How does changing the number of intervals in a partition affect the accuracy of Riemann sums?
  • Changing the number of intervals in a partition affects the accuracy of Riemann sums because more intervals generally lead to a finer partition. A finer partition means that each subinterval has a smaller width, which allows for better approximation of the area under the curve. As you increase the number of intervals, the difference between upper and lower sums decreases, making Riemann sums converge closer to the actual value of the integral.
• Discuss how different types of partitions (uniform vs. non-uniform) influence Riemann integration.
  • Uniform partitions consist of equal-width subintervals, while non-uniform partitions have varying widths. The choice between these types can impact how easily one can compute Riemann sums. Uniform partitions are often simpler for calculations but may not capture fluctuations in function behavior effectively, whereas non-uniform partitions can provide better approximations when function behavior varies significantly across the interval. This flexibility is crucial when dealing with complex functions to ensure accurate integration results.
• Evaluate how understanding partitions contributes to determining whether a function is Riemann integrable.
  • Understanding partitions is vital for determining Riemann integrability because it allows one to apply criteria like boundedness and continuity. A function is considered Riemann integrable if its upper and lower sums converge to the same limit as partitions become finer. By analyzing how various partitions affect Riemann sums, one can deduce whether specific properties like discontinuities hinder or allow for integration. Therefore, knowing how to manipulate and evaluate partitions aids in confirming if a function meets integrability requirements.

Related terms

Riemann Sum: A method for approximating the value of a definite integral by summing the products of function values at chosen points and the widths of corresponding subintervals.

Upper and Lower Sums: These are two types of sums used in Riemann integration that provide upper and lower estimates of the area under a curve based on chosen partitions.

Integrable Function: A function that can be integrated over an interval, meaning that its Riemann sums converge to a limit as the partition gets finer.