Riemann Sums

Definition

Riemann sums are a method of approximating the area under a curve by dividing it into smaller subintervals, calculating the sum of the areas of rectangles formed over these intervals. This technique is foundational for understanding integration, as it helps to establish the concept of Riemann integrability and provides insight into how functions behave over intervals, particularly in relation to their continuity and boundedness.

5 Must Know Facts For Your Next Test

1. Riemann sums can be classified as left-hand, right-hand, or midpoint sums, depending on how you choose the sample points within each subinterval.
2. A function is Riemann integrable on a closed interval if its set of discontinuities has measure zero, meaning it does not significantly affect the area approximation.
3. As the number of subintervals increases and their widths decrease, Riemann sums converge to the exact value of the definite integral.
4. The choice of partition affects the accuracy of Riemann sums; finer partitions generally yield better approximations.
5. Riemann sums provide a practical way to understand more complex concepts in calculus, especially when moving toward the Fundamental Theorem of Calculus.

Review Questions

• How do different types of Riemann sums (left-hand, right-hand, and midpoint) affect the approximation of areas under curves?
  • Different types of Riemann sums yield varying approximations for the area under curves due to their sampling points within each subinterval. Left-hand sums use the left endpoint for each rectangle's height, potentially underestimating areas when functions are increasing. Right-hand sums use the right endpoint, which may overestimate areas under increasing functions. Midpoint sums provide a balance by sampling at the midpoint, often yielding better approximations than either left or right-hand methods. The choice influences how close you get to the actual area as partitions become finer.
• Discuss how Riemann sums relate to Riemann integrability and what conditions must be met for a function to be considered Riemann integrable.
  • Riemann sums are integral in determining whether a function is Riemann integrable because they approximate the area under a curve through partitions. A function is considered Riemann integrable if it is bounded on a closed interval and its set of discontinuities has measure zero. This means that even with some discontinuities, if they are minimal in terms of their 'size', they won’t affect the overall area approximation significantly. Thus, Riemann sums serve as both a tool for approximating areas and a criterion for assessing integrability.
• Evaluate how Riemann sums provide insight into understanding continuity and boundedness of functions in relation to integration.
  • Riemann sums illustrate key aspects of continuity and boundedness by showing how these properties influence area calculations. For continuous functions on closed intervals, Riemann sums converge neatly to their definite integrals as partitions refine. In contrast, if a function is unbounded or has significant discontinuities, the estimates from Riemann sums can vary greatly or fail to converge to a meaningful limit. Understanding these connections emphasizes why continuity and boundedness are crucial for integration: they ensure that area approximations become increasingly accurate as we partition intervals more finely.

Related terms

Definite Integral: The definite integral is the limit of Riemann sums as the width of the subintervals approaches zero, representing the exact area under a curve.

Partition: A partition is a division of the interval into smaller subintervals, which is essential for calculating Riemann sums.

Upper and Lower Sums: Upper and lower sums are estimates of the area under a curve based on the maximum and minimum function values within each subinterval, providing bounds for the Riemann sum.