Subinterval

Definition

A subinterval is a smaller interval contained within a larger interval. It is a fundamental concept in calculus, particularly in the context of approximating areas under a curve using techniques like Riemann sums and Reimann integration.

5 Must Know Facts For Your Next Test

1. Subintervals are used in the process of approximating the area under a curve by dividing the interval of interest into smaller, more manageable parts.
2. The accuracy of area approximation using Riemann sums increases as the number of subintervals increases and their widths decrease.
3. The width of each subinterval is calculated by dividing the total interval length by the number of subintervals.
4. The choice of endpoints within each subinterval, such as the left, right, or midpoint, determines the type of Riemann sum (left-hand, right-hand, or midpoint, respectively).
5. Subintervals play a crucial role in the definition and calculation of the definite integral, as the limit of Riemann sums over increasingly fine partitions of the interval leads to the integral value.

Review Questions

• Explain how the concept of subintervals is used in the approximation of areas under a curve.
  • The concept of subintervals is central to the approximation of areas under a curve using techniques like Riemann sums. By dividing the interval of interest into smaller, more manageable subintervals, the area can be estimated as the sum of the areas of the rectangles formed by the subintervals. The accuracy of this approximation increases as the number of subintervals grows and their widths decrease, allowing the rectangles to better capture the shape of the curve.
• Describe how the choice of endpoints within each subinterval affects the type of Riemann sum used to approximate the area.
  • The choice of endpoints within each subinterval determines the type of Riemann sum used to approximate the area under a curve. If the left endpoint of each subinterval is used, the resulting approximation is called a left-hand Riemann sum. If the right endpoint is used, it is a right-hand Riemann sum. If the midpoint of each subinterval is used, it is a midpoint Riemann sum. The type of Riemann sum chosen can impact the accuracy of the area approximation, with midpoint Riemann sums generally providing the most accurate results.
• Explain how the concept of subintervals is related to the definition and calculation of the definite integral.
  • The concept of subintervals is fundamental to the definition and calculation of the definite integral. The definite integral is defined as the limit of Riemann sums as the number of subintervals approaches infinity and the width of each subinterval approaches zero. By dividing the interval of interest into an increasingly large number of smaller subintervals, the Riemann sum approximation becomes more accurate and approaches the true value of the definite integral. This relationship between subintervals and the definite integral is a crucial aspect of the Fundamental Theorem of Calculus, which connects the integral and derivative operations.

Related terms

Interval: An interval is a set of real numbers that lie between two given values, including or excluding the endpoints.

Partition:

A partition is a division of an interval into a finite number of subintervals, typically of equal length.

Riemann Sum:

A Riemann sum is an approximation of the area under a curve by dividing the interval into subintervals and summing the areas of the rectangles formed by the subintervals.