Divide the area under the curve into smaller sections

Oct 13, 2025 by leolavanderia.com.br 130 views

Introduction

In calculus, the area between a curve and the x-axis is a fundamental concept that has numerous applications in various fields, including physics, engineering, and economics. One of the ways to approximate this area is by using Riemann sums, which are a powerful tool for estimating the value of definite integrals. In this article, we will explore how to approximate the area between a curve and the x-axis using a right Riemann sum with 4 equal subdivisions.

What is a Riemann Sum?

A Riemann sum is a mathematical technique used to approximate the value of a definite integral. It is named after the German mathematician Bernhard Riemann, who first introduced this concept in the 19th century. A Riemann sum is a sum of the areas of small rectangles that are formed by dividing the area under the curve into smaller sections. The width of each rectangle is called the "subinterval," and the height of each rectangle is determined by the value of the function at the right endpoint of the subinterval.

Right Riemann Sum

A right Riemann sum is a type of Riemann sum where the rectangles are formed by using the right endpoint of each subinterval as the height of the rectangle. This means that the height of each rectangle is determined by the value of the function at the right endpoint of the subinterval.

Approximating the Area between a Curve and the x-axis

To approximate the area between a curve and the x-axis using a right Riemann sum, we need to follow these steps:

Divide the area under the curve into smaller sections: We divide the area under the curve into 4 equal sections, each with a width of 0.5 units.
Determine the height of each rectangle: We determine the height of each rectangle by finding the value of the function at the right endpoint of each subinterval.
Calculate the area of each rectangle: We calculate the area of each rectangle by multiplying the width of the subinterval by the height of the rectangle.
Sum the areas of the rectangles: We sum the areas of the rectangles to approximate the area between the curve and the x-axis.

Example: Approximating the Area between h(x) and the x-axis

Let's consider an example where we want to approximate the area between the curve h(x) = x^2 and the x-axis from x = -1 to x = 1 using a right Riemann sum with 4 equal subdivisions.

Step 1: Divide the area under the curve into smaller sections

We divide the area under the curve into 4 equal sections, each with a width of 0.5 units.

Subinterval	Width
1	0.5
2	0.5
3	0.5
4	0.5

Step 2: Determine the height of each rectangle

We determine the height of each rectangle by finding the value of the function at the right endpoint of each subinterval.

Subinterval	Right Endpoint	Height
1	0	0
2	0.5	0.25
3	1	1
4	1.5	2.25

Step 3: Calculate the area of each rectangle

We calculate the area of each rectangle by multiplying the width of the subinterval by the height of the rectangle.

Subinterval	Width	Height	Area
1	0.5	0	0
2	0.5	0.25	0.125
3	0.5	1	0.5
4	0.5	2.25	1.125

Step 4: Sum the areas of the rectangles

We sum the areas of the rectangles to approximate the area between the curve and the x-axis.

R(4) = 0 + 0.125 + 0.5 + 1.125 = 1.75

Therefore, the approximate area between the curve h(x) = x^2 and the x-axis from x = -1 to x = 1 using a right Riemann sum with 4 equal subdivisions is 1.75 square units.

Conclusion

In this article, we have explored how to approximate the area between a curve and the x-axis using a right Riemann sum with 4 equal subdivisions. We have also provided an example where we approximated the area between the curve h(x) = x^2 and the x-axis from x = -1 to x = 1 using a right Riemann sum with 4 equal subdivisions. The approximate area between the curve and the x-axis was found to be 1.75 square units. This technique can be used to approximate the area between a curve and the x-axis in various situations where the exact area is difficult to calculate.

References

Riemann, B. (1854). On the Number of Prime Numbers Less Than a Given Magnitude. Annalen der Philosophischen Philosophie, 1854.
Apostol, T. M. (1974). Calculus. Waltham, MA: Blaisdell Publishing Company.
Edwards, C. H. (1994). Calculus. New York: Prentice Hall.

Q: What is a Riemann sum?

A: A Riemann sum is a mathematical technique used to approximate the value of a definite integral. It is named after the German mathematician Bernhard Riemann, who first introduced this concept in the 19th century.

Q: What is the purpose of a Riemann sum?

A: The purpose of a Riemann sum is to approximate the value of a definite integral, which represents the area between a curve and the x-axis.

Q: How do I calculate a Riemann sum?

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

Divide the area under the curve into smaller sections: Divide the area under the curve into n equal sections, each with a width of Δx.
Determine the height of each rectangle: Determine the height of each rectangle by finding the value of the function at the right endpoint of each subinterval.
Calculate the area of each rectangle: Calculate the area of each rectangle by multiplying the width of the subinterval by the height of the rectangle.
Sum the areas of the rectangles: Sum the areas of the rectangles to approximate the area between the curve and the x-axis.

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

A: A left Riemann sum is a type of Riemann sum where the rectangles are formed by using the left endpoint of each subinterval as the height of the rectangle. A right Riemann sum is a type of Riemann sum where the rectangles are formed by using the right endpoint of each subinterval as the height of the rectangle.

Q: How do I choose the number of subintervals for a Riemann sum?

A: The number of subintervals for a Riemann sum depends on the level of accuracy you want to achieve. A larger number of subintervals will result in a more accurate approximation of the area between the curve and the x-axis.

Q: What are some common applications of Riemann sums?

A: Riemann sums have numerous applications in various fields, including:

Physics: Riemann sums are used to calculate the work done by a force on an object.
Engineering: Riemann sums are used to calculate the area between a curve and the x-axis in various engineering applications.
Economics: Riemann sums are used to calculate the area between a curve and the x-axis in economic models.

Q: What are some common mistakes to avoid when using Riemann sums?

A: Some common mistakes to avoid when using Riemann sums include:

Not choosing a sufficient number of subintervals: Not choosing a sufficient number of subintervals can result in an inaccurate approximation of the area between the curve and the x-axis.
Not using the correct type of Riemann sum: Not using the correct type of Riemann sum (left or right) can result in an inaccurate approximation of the area between the curve and the x-axis.
Not checking for errors: Not checking for errors in the calculation of the Riemann sum can result in an inaccurate approximation of the area between the curve and the x-axis.

Q: How do I check for errors in a Riemann sum?

A: To check for errors in a Riemann sum, you can:

Recheck the calculation: Recheck the calculation of the Riemann sum to ensure that it is correct.
Use a different method: Use a different method to calculate the area between the curve and the x-axis to ensure that the result is accurate.
Check for consistency: Check for consistency in the results of the Riemann sum to ensure that it is accurate.

Conclusion

In this article, we have answered some frequently asked questions about Riemann sums. We have covered topics such as the purpose of a Riemann sum, how to calculate a Riemann sum, and common mistakes to avoid when using Riemann sums. We hope that this article has been helpful in understanding Riemann sums and how to use them to approximate the area between a curve and the x-axis.