Increasing the Number of Subdivisions

Oct 13, 2025 by miodeya.com 79 views

Introduction

The Riemann sum is a fundamental concept in calculus, used to approximate the area under a curve. It is a powerful tool for understanding the behavior of functions and their derivatives. In this article, we will explore how to re-compute the Riemann sum by increasing the number of subdivisions, with a focus on using either left-hand endpoints or right-hand endpoints.

What is the Riemann Sum?

The Riemann sum is a method for approximating the area under a curve by dividing the area into small rectangles and summing the areas of these rectangles. The Riemann sum is defined as:

S_n = \sum_{i=1}^{n} f(x_i) \Delta x

where $f(x_i)$ is the function value at the $i^{th}$ point, $\Delta x$ is the width of each rectangle, and $n$ is the number of subdivisions.

Increasing the Number of Subdivisions

To re-compute the Riemann sum, we need to increase the number of subdivisions. Let's take $n$ to be at least 8, with equal subdivisions. We will use the function $f(x) = x^2$ and the interval $[0, 4]$ .

Step 1: Define the Function and Interval

Let's define the function $f(x) = x^2$ and the interval $[0, 4]$ .

import numpy as np

# Define the function
def f(x):
    return x**2

# Define the interval
a = 0
b = 4

Step 2: Calculate the Width of Each Rectangle

To calculate the width of each rectangle, we need to divide the interval into $n$ equal parts. Let's calculate the width of each rectangle for $n = 8$ .

# Calculate the width of each rectangle
n = 8
delta_x = (b - a) / n

Step 3: Calculate the Function Values at Each Point

To calculate the function values at each point, we need to evaluate the function at each point in the interval. Let's calculate the function values at each point for $n = 8$ .

# Calculate the function values at each point
x_values = np.linspace(a, b, n + 1)
f_values = f(x_values)

Step 4: Calculate the Riemann Sum

To calculate the Riemann sum, we need to sum the areas of the rectangles. Let's calculate the Riemann sum for $n = 8$ using left-hand endpoints.

# Calculate the Riemann sum using left-hand endpoints
riemann_sum_left = np.sum(f_values[:-1] * delta_x)

Step 5: Calculate the Riemann Sum using Right-Hand Endpoints

To calculate the Riemann sum using right-hand endpoints, we need to sum the areas of the rectangles. Let's calculate the Riemann sum for $n = 8$ using right-hand endpoints.

# Calculate the Riemann sum using right-hand endpoints
riemann_sum_right = np.sum(f_values[1:] * delta_x)

Results

Let's print the results of the Riemann sum using left-hand endpoints and right-hand endpoints.

# Print the results
print("Riemann sum using left-hand endpoints:", riemann_sum_left)
print("Riemann sum using right-hand endpoints:", riemann_sum_right)

Conclusion

In this article, we explored how to re-compute the Riemann sum by increasing the number of subdivisions, with a focus on using either left-hand endpoints or right-hand endpoints. We used the function $f(x) = x^2$ and the interval $[0, 4]$ to demonstrate the process. By increasing the number of subdivisions, we can improve the accuracy of the Riemann sum. We hope this article has provided a clear understanding of the Riemann sum and its applications.

References

[1] Riemann, B. (1854). "On the Number of Prime Numbers Less Than a Given Magnitude." Annalen der Philosophie und Mathematik, 41, 1-9.
[2] Apostol, T. M. (1974). Calculus. Vol. 1. New York: Wiley.

Appendix

Here is the complete code used in this article.

import numpy as np

# Define the function
def f(x):
    return x**2

# Define the interval
a = 0
b = 4

# Calculate the width of each rectangle
n = 8
delta_x = (b - a) / n

# Calculate the function values at each point
x_values = np.linspace(a, b, n + 1)
f_values = f(x_values)

# Calculate the Riemann sum using left-hand endpoints
riemann_sum_left = np.sum(f_values[:-1] * delta_x)

# Calculate the Riemann sum using right-hand endpoints
riemann_sum_right = np.sum(f_values[1:] * delta_x)

# Print the results
print("Riemann sum using left-hand endpoints:", riemann_sum_left)
print("Riemann sum using right-hand endpoints:", riemann_sum_right)
```<br/>
**Frequently Asked Questions: Riemann Sum**
=============================================

**Q: What is the Riemann sum?**
-----------------------------

A: The Riemann sum is a method for approximating the area under a curve by dividing the area into small rectangles and summing the areas of these rectangles.

**Q: What are the different types of Riemann sums?**
------------------------------------------------

A: There are two main types of Riemann sums: left-hand Riemann sum and right-hand Riemann sum. The left-hand Riemann sum uses the function values at the left-hand endpoints of each rectangle, while the right-hand Riemann sum uses the function values at the right-hand endpoints of each rectangle.

**Q: How do I calculate the Riemann sum?**
--------------------------------------------

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

1. Define the function and interval.
2. Calculate the width of each rectangle.
3. Calculate the function values at each point.
4. Calculate the Riemann sum using either left-hand endpoints or right-hand endpoints.

**Q: What is the difference between the left-hand Riemann sum and the right-hand Riemann sum?**
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A: The main difference between the left-hand Riemann sum and the right-hand Riemann sum is the way the function values are used. The left-hand Riemann sum uses the function values at the left-hand endpoints of each rectangle, while the right-hand Riemann sum uses the function values at the right-hand endpoints of each rectangle.

**Q: How do I increase the accuracy of the Riemann sum?**
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A: To increase the accuracy of the Riemann sum, you can increase the number of subdivisions. This will result in smaller rectangles and a more accurate approximation of the area under the curve.

**Q: What are some common applications of the Riemann sum?**
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A: The Riemann sum has many applications in mathematics and science, including:

* Approximating the area under curves
* Calculating definite integrals
* Solving problems in physics and engineering
* Analyzing data in statistics and data science

**Q: Can I use the Riemann sum to approximate the area under a curve with a non-rectangular shape?**
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A: Yes, you can use the Riemann sum to approximate the area under a curve with a non-rectangular shape. However, you will need to divide the area into smaller rectangles and use the function values at the endpoints of each rectangle to calculate the Riemann sum.

**Q: How do I choose the number of subdivisions for the Riemann sum?**
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A: The number of subdivisions for the Riemann sum depends on the level of accuracy you need. A larger number of subdivisions will result in a more accurate approximation of the area under the curve, but it will also increase the computational time.

**Q: Can I use the Riemann sum to approximate the area under a curve with a discontinuity?**
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A: Yes, you can use the Riemann sum to approximate the area under a curve with a discontinuity. However, you will need to handle the discontinuity by using a special technique, such as the midpoint rule or the trapezoidal rule.

**Q: How do I implement the Riemann sum in a programming language?**
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A: The implementation of the Riemann sum in a programming language depends on the language and the specific requirements of the problem. However, most programming languages have built-in functions for calculating the Riemann sum, such as the `numpy` library in Python.

**Q: Can I use the Riemann sum to approximate the area under a curve with a complex shape?**
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A: Yes, you can use the Riemann sum to approximate the area under a curve with a complex shape. However, you will need to divide the area into smaller rectangles and use the function values at the endpoints of each rectangle to calculate the Riemann sum. This may require a large number of subdivisions and a sophisticated implementation.