Understanding the Left Riemann Sum

Oct 13, 2025 by miodeya.com 148 views

Introduction

In calculus, the Riemann sum is a method used to approximate the value of a definite integral. It is a fundamental concept in understanding the relationship between the area under a curve and the definite integral. In this article, we will explore the left Riemann sum approximation for the definite integral of a function f(x) from 2 to 14 using 5 subintervals.

Understanding the Left Riemann Sum

The left Riemann sum is a type of Riemann sum that approximates the area under a curve by dividing the area into small rectangles and summing the areas of these rectangles. The left Riemann sum is calculated by using the left endpoint of each subinterval to determine the height of the rectangle.

Calculating the Left Riemann Sum

To calculate the left Riemann sum, we need to follow these steps:

Divide the interval: Divide the interval [2, 14] into 5 equal subintervals. Each subinterval will have a width of (14 - 2) / 5 = 2.2.
Find the left endpoint: Find the left endpoint of each subinterval. The left endpoints are 2, 4, 6, 8, and 10.
Find the function value: Find the value of the function f(x) at each left endpoint. The function values are shown in the table below.

x	f(x)
2	3
4	5
6	7
8	9
10	11

Calculate the area: Calculate the area of each rectangle by multiplying the function value at the left endpoint by the width of the subinterval.
Sum the areas: Sum the areas of all the rectangles to get the left Riemann sum approximation.

Calculating the Left Riemann Sum Approximation

Using the values from the table, we can calculate the left Riemann sum approximation as follows:

For the first subinterval [2, 4], the area is 3 * 2.2 = 6.6.
For the second subinterval [4, 6], the area is 5 * 2.2 = 11.
For the third subinterval [6, 8], the area is 7 * 2.2 = 15.4.
For the fourth subinterval [8, 10], the area is 9 * 2.2 = 19.8.
For the fifth subinterval [10, 14], the area is 11 * 4.2 = 46.2.

The left Riemann sum approximation is the sum of these areas: 6.6 + 11 + 15.4 + 19.8 + 46.2 = 99.

Conclusion

In this article, we have explored the left Riemann sum approximation for the definite integral of a function f(x) from 2 to 14 using 5 subintervals. We have calculated the left Riemann sum approximation using the values from the table and found that it is approximately 99.

Discussion

The left Riemann sum approximation is a useful tool for approximating the value of a definite integral. However, it is not always accurate, especially for functions with a high degree of curvature. In such cases, the midpoint Riemann sum or the trapezoidal rule may be more accurate.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart

Table of Values

x	f(x)
2	3
4	5
6	7
8	9
10	11

Left Riemann Sum Approximation

Introduction

In our previous article, we explored the left Riemann sum approximation for the definite integral of a function f(x) from 2 to 14 using 5 subintervals. In this article, we will answer some frequently asked questions about the left Riemann sum approximation.

Q: What is the left Riemann sum approximation?

A: The left Riemann sum approximation is a method used to approximate the value of a definite integral by dividing the area under a curve into small rectangles and summing the areas of these rectangles.

Q: How is the left Riemann sum approximation calculated?

A: To calculate the left Riemann sum approximation, you need to follow these steps:

Divide the interval: Divide the interval [a, b] into n equal subintervals. Each subinterval will have a width of (b - a) / n.
Find the left endpoint: Find the left endpoint of each subinterval.
Find the function value: Find the value of the function f(x) at each left endpoint.
Calculate the area: Calculate the area of each rectangle by multiplying the function value at the left endpoint by the width of the subinterval.
Sum the areas: Sum the areas of all the rectangles to get the left Riemann sum approximation.

Q: What is the difference between the left Riemann sum and the right Riemann sum?

A: The left Riemann sum and the right Riemann sum are both methods used to approximate the value of a definite integral. The main difference between them is that the left Riemann sum uses the left endpoint of each subinterval to determine the height of the rectangle, while the right Riemann sum uses the right endpoint.

Q: When is the left Riemann sum approximation accurate?

A: The left Riemann sum approximation is accurate when the function f(x) is continuous and has a small degree of curvature. However, it may not be accurate for functions with a high degree of curvature.

Q: Can the left Riemann sum approximation be used for functions with a high degree of curvature?

A: No, the left Riemann sum approximation is not suitable for functions with a high degree of curvature. In such cases, the midpoint Riemann sum or the trapezoidal rule may be more accurate.

Q: How can the accuracy of the left Riemann sum approximation be improved?

A: The accuracy of the left Riemann sum approximation can be improved by:

Increasing the number of subintervals: Increasing the number of subintervals will result in a more accurate approximation.
Using a smaller width: Using a smaller width for each subinterval will result in a more accurate approximation.
Using a more accurate method: Using a more accurate method, such as the midpoint Riemann sum or the trapezoidal rule, may result in a more accurate approximation.

Conclusion

In this article, we have answered some frequently asked questions about the left Riemann sum approximation. We hope that this article has provided you with a better understanding of the left Riemann sum approximation and how it can be used to approximate the value of a definite integral.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart

Table of Values

x	f(x)
2	3
4	5
6	7
8	9
10	11

Left Riemann Sum Approximation

The left Riemann sum approximation is approximately 99.

Frequently Asked Questions

Q: What is the left Riemann sum approximation? A: The left Riemann sum approximation is a method used to approximate the value of a definite integral by dividing the area under a curve into small rectangles and summing the areas of these rectangles.
Q: How is the left Riemann sum approximation calculated? A: To calculate the left Riemann sum approximation, you need to follow these steps:
1. Divide the interval: Divide the interval [a, b] into n equal subintervals. Each subinterval will have a width of (b - a) / n.
2. Find the left endpoint: Find the left endpoint of each subinterval.
3. Find the function value: Find the value of the function f(x) at each left endpoint.
4. Calculate the area: Calculate the area of each rectangle by multiplying the function value at the left endpoint by the width of the subinterval.
5. Sum the areas: Sum the areas of all the rectangles to get the left Riemann sum approximation.
Q: What is the difference between the left Riemann sum and the right Riemann sum? A: The left Riemann sum and the right Riemann sum are both methods used to approximate the value of a definite integral. The main difference between them is that the left Riemann sum uses the left endpoint of each subinterval to determine the height of the rectangle, while the right Riemann sum uses the right endpoint.