The Limit Definition of a Definite Integral

Oct 13, 2025 by windailys.com 61 views

Introduction

The definite integral of a function f(x) from a to b is a fundamental concept in calculus, representing the area under the curve of the function between the limits of a and b. In this article, we will explore the relationship between the Riemann sum and the limit definition of a definite integral, and determine the value of Δxi.

The Riemann Sum

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

∑[f(x_i) * Δx_i] from i=1 to n

where f(x_i) is the value of the function at the i-th point, Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

The Limit Definition of a Definite Integral

The limit definition of a definite integral is a mathematical definition of the definite integral as a limit of a Riemann sum. It is defined as:

∫[a,b] f(x) dx = lim(Δx_i → 0) ∑[f(x_i) * Δx_i] from i=1 to n

where Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

Determining the Value of Δxi

To determine the value of Δxi, we need to consider the relationship between the Riemann sum and the limit definition of a definite integral. As the width of the rectangles (Δxi) approaches zero, the Riemann sum approaches the definite integral.

Let's consider a function f(x) = x^2, and we want to find the definite integral of f(x) from -2 to 3. We can divide the area under the curve into small rectangles, and sum the areas of the rectangles.

Calculating the Riemann Sum

To calculate the Riemann sum, we need to divide the area under the curve into small rectangles. Let's divide the area into 10 rectangles, each with a width of Δx = (3 - (-2))/10 = 0.5.

x_i	f(x_i)	Δx_i	f(x_i) * Δx_i
-1.5	2.25	0.5	1.125
-1	1	0.5	0.5
-0.5	0.25	0.5	0.125
0	0	0.5	0
0.5	0.25	0.5	0.125
1	1	0.5	0.5
1.5	2.25	0.5	1.125
2	4	0.5	2
2.5	6.25	0.5	3.125
3	9	0.5	4.5

Calculating the Limit

To calculate the limit, we need to sum the areas of the rectangles and divide by the number of rectangles.

∑[f(x_i) * Δx_i] from i=1 to n = 1.125 + 0.5 + 0.125 + 0 + 0.125 + 0.5 + 1.125 + 2 + 3.125 + 4.5 = 13.5

Now, we need to divide the sum by the number of rectangles (n = 10) and multiply by the width of the rectangles (Δx = 0.5).

lim(Δx_i → 0) ∑[f(x_i) * Δx_i] from i=1 to n = (13.5 * 0.5) / 10 = 0.675

Conclusion

In conclusion, the value of Δxi is 0.5, which is the width of the rectangles used to approximate the area under the curve of the function f(x) = x^2 from -2 to 3. As the width of the rectangles approaches zero, the Riemann sum approaches the definite integral.

The Importance of Δxi

The value of Δxi is crucial in the calculation of the definite integral. As the width of the rectangles approaches zero, the Riemann sum approaches the definite integral. This is because the Riemann sum is an approximation of the definite integral, and the accuracy of the approximation depends on the width of the rectangles.

Future Work

In future work, we can explore other methods of approximating the definite integral, such as the trapezoidal rule and Simpson's rule. We can also investigate the relationship between the Riemann sum and other mathematical concepts, such as the fundamental theorem of calculus.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "The Riemann Sum" by Wolfram MathWorld

Glossary

Riemann sum: a method of approximating the area under a curve by dividing the area into small rectangles and summing the areas of the rectangles.
Limit definition of a definite integral: a mathematical definition of the definite integral as a limit of a Riemann sum.
Δxi: the width of the i-th rectangle in the Riemann sum.
n: the number of rectangles in the Riemann sum.
Q&A: The Riemann Sum and the Limit Definition of a Definite Integral ====================================================================

Q: What is the Riemann sum?

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

Q: What is the limit definition of a definite integral?

A: The limit definition of a definite integral is a mathematical definition of the definite integral as a limit of a Riemann sum.

Q: What is the relationship between the Riemann sum and the limit definition of a definite integral?

A: The Riemann sum is an approximation of the definite integral, and the accuracy of the approximation depends on the width of the rectangles. As the width of the rectangles approaches zero, the Riemann sum approaches the definite integral.

Q: How do you calculate the Riemann sum?

A: To calculate the Riemann sum, you need to divide the area under the curve into small rectangles, and sum the areas of the rectangles. The formula for the Riemann sum is:

∑[f(x_i) * Δx_i] from i=1 to n

where f(x_i) is the value of the function at the i-th point, Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

Q: What is the value of Δxi?

A: The value of Δxi is the width of the i-th rectangle in the Riemann sum.

Q: How do you determine the value of Δxi?

A: To determine the value of Δxi, you need to consider the relationship between the Riemann sum and the limit definition of a definite integral. As the width of the rectangles approaches zero, the Riemann sum approaches the definite integral.

Q: What is the importance of Δxi?

A: The value of Δxi is crucial in the calculation of the definite integral. As the width of the rectangles approaches zero, the Riemann sum approaches the definite integral.

Q: Can you give an example of how to calculate the Riemann sum?

A: Yes, let's consider a function f(x) = x^2, and we want to find the definite integral of f(x) from -2 to 3. We can divide the area under the curve into small rectangles, and sum the areas of the rectangles.

x_i	f(x_i)	Δx_i	f(x_i) * Δx_i
-1.5	2.25	0.5	1.125
-1	1	0.5	0.5
-0.5	0.25	0.5	0.125
0	0	0.5	0
0.5	0.25	0.5	0.125
1	1	0.5	0.5
1.5	2.25	0.5	1.125
2	4	0.5	2
2.5	6.25	0.5	3.125
3	9	0.5	4.5

Q: Can you explain the concept of the limit definition of a definite integral?

A: Yes, the limit definition of a definite integral is a mathematical definition of the definite integral as a limit of a Riemann sum. It is defined as:

∫[a,b] f(x) dx = lim(Δx_i → 0) ∑[f(x_i) * Δx_i] from i=1 to n

where Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

Q: What is the significance of the limit definition of a definite integral?

A: The limit definition of a definite integral is significant because it provides a rigorous mathematical definition of the definite integral. It also provides a way to calculate the definite integral using the Riemann sum.

Q: Can you give an example of how to use the limit definition of a definite integral?

A: Yes, let's consider a function f(x) = x^2, and we want to find the definite integral of f(x) from -2 to 3. We can use the limit definition of a definite integral to calculate the definite integral.

∫[-2,3] x^2 dx = lim(Δx_i → 0) ∑[x_i^2 * Δx_i] from i=1 to n

where Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

Q: What is the relationship between the Riemann sum and the fundamental theorem of calculus?

A: The Riemann sum and the fundamental theorem of calculus are related in that the Riemann sum is an approximation of the definite integral, and the fundamental theorem of calculus provides a way to calculate the definite integral using the Riemann sum.

Q: Can you explain the concept of the fundamental theorem of calculus?

A: Yes, the fundamental theorem of calculus is a theorem that states that the definite integral of a function f(x) from a to b is equal to the antiderivative of f(x) evaluated at b minus the antiderivative of f(x) evaluated at a.

∫[a,b] f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

Q: What is the significance of the fundamental theorem of calculus?

A: The fundamental theorem of calculus is significant because it provides a way to calculate the definite integral using the antiderivative of the function. It also provides a way to calculate the definite integral using the Riemann sum.

Q: Can you give an example of how to use the fundamental theorem of calculus?

A: Yes, let's consider a function f(x) = x^2, and we want to find the definite integral of f(x) from -2 to 3. We can use the fundamental theorem of calculus to calculate the definite integral.

∫[-2,3] x^2 dx = F(3) - F(-2)

where F(x) is the antiderivative of x^2.

F(x) = (1/3)x^3

F(3) = (1/3)(3)^3 = 9

F(-2) = (1/3)(-2)^3 = -8/3

∫[-2,3] x^2 dx = 9 - (-8/3) = 35/3

Q: What is the relationship between the Riemann sum and the definite integral?

A: The Riemann sum and the definite integral are related in that the Riemann sum is an approximation of the definite integral. As the width of the rectangles approaches zero, the Riemann sum approaches the definite integral.

Q: Can you explain the concept of the definite integral?

A: Yes, the definite integral is a mathematical concept that represents the area under a curve between two points. It is defined as:

∫[a,b] f(x) dx = lim(Δx_i → 0) ∑[f(x_i) * Δx_i] from i=1 to n

where Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

Q: What is the significance of the definite integral?

A: The definite integral is significant because it provides a way to calculate the area under a curve between two points. It is also used in many applications, such as physics, engineering, and economics.

Q: Can you give an example of how to use the definite integral?

A: Yes, let's consider a function f(x) = x^2, and we want to find the definite integral of f(x) from -2 to 3. We can use the definite integral to calculate the area under the curve between -2 and 3.

∫[-2,3] x^2 dx = lim(Δx_i → 0) ∑[x_i^2 * Δx_i] from i=1 to n

where Δx_i is the width of the i-th rectangle, and n is the number of rectangles.

Q: What is the relationship between the definite integral and the fundamental theorem of calculus?

A: The definite integral and the fundamental theorem of calculus are related in that the definite integral is used in the fundamental theorem of calculus to calculate the definite integral.

Q: Can you explain the concept of the fundamental theorem of calculus?

∫[a,b] f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

Q: What is the significance of the fundamental theorem of calculus?

A: The fundamental theorem of calculus is significant because it provides a way to calculate the definite integral using the antiderivative of the