Riemann Integrable

Definition

A function is called Riemann integrable if it can be approximated by Riemann sums over its interval, and if the limit of these sums exists as the partition of the interval becomes finer. This concept is central to understanding how to calculate the area under a curve using limits and partitions, emphasizing that a Riemann integrable function must be bounded and its set of discontinuities must have measure zero.

5 Must Know Facts For Your Next Test

1. Riemann integrability requires the function to be bounded on a closed interval.
2. A key condition for Riemann integrability is that the set of discontinuities of the function must have measure zero.
3. Continuous functions on closed intervals are always Riemann integrable.
4. If a function is Riemann integrable, then it can be approximated as closely as desired by a Riemann sum with sufficiently small partition intervals.
5. The concept of being Riemann integrable serves as a foundation for more advanced integration techniques like Lebesgue integration.

Review Questions

• What conditions must be met for a function to be considered Riemann integrable, and why are these conditions significant?
  • For a function to be considered Riemann integrable, it must be bounded on a closed interval, and its set of discontinuities must have measure zero. These conditions are significant because they ensure that the area under the curve can be accurately approximated using Riemann sums. If either condition fails, it can lead to infinite discontinuities or unbounded behavior that prevents reliable approximation of the integral.
• Compare Riemann sums with other methods of integration in terms of their effectiveness in approximating areas under curves.
  • Riemann sums offer a straightforward approach for approximating the area under curves using partitions of an interval, but they can struggle with functions that have many discontinuities or are unbounded. In contrast, methods like the trapezoidal rule or Simpson's rule refine this process by using average heights or parabolic sections to improve accuracy. These methods often yield better results for functions that are not well-behaved, illustrating that while Riemann sums provide foundational understanding, they may not always be the most effective choice for complex functions.
• Evaluate how the concept of measure zero impacts the understanding of Riemann integrability and its limitations in defining integrals for more complex functions.
  • The concept of measure zero significantly impacts our understanding of Riemann integrability by allowing functions with sets of discontinuities that are 'small' in a sense to still be integrated. However, this creates limitations since there are functions with dense sets of discontinuities that cannot be expressed as Riemann integrable despite being bounded. This recognition leads to the development of more robust integration theories, such as Lebesgue integration, which can handle more complicated functions and provide greater flexibility in analyzing convergence and continuity.

Related terms

Riemann Sum: A Riemann sum is a method for approximating the total area under a curve by dividing the region into smaller segments, calculating the area of rectangles formed over these segments, and summing these areas.

Partition: A partition of an interval is a finite set of points that divides the interval into subintervals, used in defining Riemann sums and evaluating the Riemann integral.

Measure Zero: A set has measure zero if its total length (or volume in higher dimensions) can be made arbitrarily small, which means it does not contribute to the area when considering integrability.