Riemann Integral

Definition

The Riemann integral is a method of assigning a number to the area under a curve defined by a function over a specified interval. This integral is calculated using the concept of partitioning the interval into subintervals, determining the function's value at specific points, and summing the resulting areas of rectangles formed, which approaches the actual area as the partition gets finer. Understanding the Riemann integral is crucial in analyzing measurable functions and their properties in integration.

5 Must Know Facts For Your Next Test

1. A function is Riemann integrable if it is continuous almost everywhere on a closed interval and has a finite number of discontinuities.
2. The Riemann integral can be expressed as $$ ext{R}(f) = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x_i$$ where $$x_i^*$$ are sample points in each subinterval.
3. The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous on an interval, its Riemann integral can be computed using its antiderivative.
4. The concept of upper and lower sums is crucial for understanding how the Riemann integral approximates the area under a curve.
5. Riemann integrals are sensitive to the choice of sample points in partitions, which can affect convergence to the actual area under the curve.

Review Questions

• How does the concept of partitions influence the calculation of the Riemann integral?
  • Partitions play a critical role in calculating the Riemann integral by dividing the interval into smaller subintervals. The finer the partition, meaning more and smaller subintervals, the more accurate the approximation becomes for calculating areas under curves. Each subinterval contributes to an approximate area based on function values at specific points, and as these partitions become infinitely small, the sum approaches the exact area represented by the integral.
• Compare and contrast the Riemann integral with the Lebesgue integral regarding their definitions and applications.
  • The Riemann integral focuses on partitioning intervals and relies on function values at specific points to approximate areas, which limits its application mainly to continuous functions. In contrast, the Lebesgue integral considers the measure of sets rather than specific intervals, allowing it to integrate a broader class of functions, including those with more complicated discontinuities. This makes the Lebesgue integral more powerful in analysis, particularly in handling functions that are not well-behaved under Riemann integration.
• Evaluate how understanding Riemann integrals contributes to grasping measurable functions and their significance in advanced mathematics.
  • Understanding Riemann integrals is foundational for comprehending measurable functions as it highlights how we quantify areas and accumulation over intervals. It introduces key concepts like continuity and boundedness essential for defining measurability. As one progresses to Lebesgue integration, this groundwork in Riemann integrals enables deeper insights into more complex scenarios involving measurable functions, emphasizing their roles in analysis and probability theory within advanced mathematical frameworks.

Related terms

Lebesgue Integral: A more general form of integration that extends the concept of integration beyond Riemann integrable functions, focusing on measuring sets rather than partitions.

Partition: A division of an interval into smaller subintervals that is used in the calculation of the Riemann integral.

Measurable Function: A function that is compatible with a measure space, meaning its pre-images of measurable sets are also measurable, allowing for the application of integration techniques.