Expressing a Definite Integral as a Riemann Sum

Oct 13, 2025 by miodeya.com 0 views

Introduction

In calculus, the definite integral is a fundamental concept that represents the area under a curve. One way to approximate this area is by using Riemann sums, which are a series of rectangles that approximate the area under the curve. In this article, we will explore how to express a definite integral as the limit of a Riemann sum.

What is a Riemann Sum?

A Riemann sum is a mathematical concept that approximates the area under a curve by dividing the area into small rectangles. The height of each rectangle is determined by the function value at a specific point, and the width of each rectangle is determined by the interval over which the function is evaluated. The Riemann sum is then calculated by summing the areas of these rectangles.

Expressing a Definite Integral as a Riemann Sum

To express a definite integral as a Riemann sum, we need to follow these steps:

Divide the interval into subintervals: Divide the interval [a, b] into n subintervals of equal width, denoted by Δx.
Choose a point in each subinterval: Choose a point in each subinterval, denoted by x_i.
Calculate the function value at each point: Calculate the function value at each point, denoted by f(x_i).
Calculate the area of each rectangle: Calculate the area of each rectangle by multiplying the function value at each point by the width of the subinterval, denoted by f(x_i) * Δx.
Sum the areas of the rectangles: Sum the areas of the rectangles to obtain the Riemann sum.

Expressing the Definite Integral as a Riemann Sum

Now, let's apply these steps to the definite integral ∫(x+5) dx from 1 to 5.

Step 1: Divide the interval into subintervals

We divide the interval [1, 5] into n subintervals of equal width, denoted by Δx. The width of each subinterval is Δx = (5 - 1) / n = 4 / n.

Step 2: Choose a point in each subinterval

We choose a point in each subinterval, denoted by x_i. Let's choose the right endpoint of each subinterval, so x_i = 1 + i * Δx.

Step 3: Calculate the function value at each point

We calculate the function value at each point, denoted by f(x_i). The function is f(x) = x + 5, so f(x_i) = x_i + 5.

Step 4: Calculate the area of each rectangle

We calculate the area of each rectangle by multiplying the function value at each point by the width of the subinterval, denoted by f(x_i) * Δx. The area of each rectangle is (x_i + 5) * Δx.

Step 5: Sum the areas of the rectangles

We sum the areas of the rectangles to obtain the Riemann sum. The Riemann sum is:

∑[(x_i + 5) * Δx] from i=1 to n

Expressing the Riemann Sum as a Limit

As the number of subintervals increases, the width of each subinterval decreases, and the Riemann sum approaches the definite integral. We can express the Riemann sum as a limit:

lim(n→∞) ∑[(x_i + 5) * Δx] from i=1 to n

Evaluating the Limit

To evaluate the limit, we can rewrite the Riemann sum as a sum of two terms:

lim(n→∞) ∑[x_i * Δx] from i=1 to n + lim(n→∞) ∑[5 * Δx] from i=1 to n

The first term is a Riemann sum of the function x, and the second term is a Riemann sum of the constant function 5.

Evaluating the First Term

The first term is a Riemann sum of the function x. We can evaluate this term by using the formula for the Riemann sum of a linear function:

∑[x_i * Δx] from i=1 to n = (b - a) * (n / 2)

In this case, a = 1, b = 5, and n is the number of subintervals. Plugging in these values, we get:

∑[x_i * Δx] from i=1 to n = (5 - 1) * (n / 2) = 2 * n

Evaluating the Second Term

The second term is a Riemann sum of the constant function 5. We can evaluate this term by using the formula for the Riemann sum of a constant function:

∑[5 * Δx] from i=1 to n = 5 * (b - a)

In this case, a = 1, b = 5, and n is the number of subintervals. Plugging in these values, we get:

∑[5 * Δx] from i=1 to n = 5 * (5 - 1) = 20

Combining the Terms

Now, we can combine the two terms to get the final result:

lim(n→∞) ∑[(x_i + 5) * Δx] from i=1 to n = lim(n→∞) [2 * n + 20]

As n approaches infinity, the term 2 * n approaches infinity, and the term 20 approaches a constant value. Therefore, the limit is:

lim(n→∞) ∑[(x_i + 5) * Δx] from i=1 to n = ∞ + 20

However, this is not the final answer. We need to evaluate the limit of the first term, which is 2 * n.

Evaluating the Limit of the First Term

The limit of the first term is:

lim(n→∞) 2 * n = ∞

However, this is not the final answer. We need to evaluate the limit of the expression (x_i + 5) * Δx.

Evaluating the Limit of the Expression

The limit of the expression (x_i + 5) * Δx is:

lim(n→∞) (x_i + 5) * Δx = lim(n→∞) (1 + i * Δx + 5) * Δx

As n approaches infinity, the width of each subinterval Δx approaches zero. Therefore, the limit is:

lim(n→∞) (1 + i * Δx + 5) * Δx = (1 + 5) * 0 = 0

However, this is not the final answer. We need to evaluate the limit of the sum of the expressions.

Evaluating the Limit of the Sum

The limit of the sum is:

lim(n→∞) ∑[(x_i + 5) * Δx] from i=1 to n = lim(n→∞) ∑[(1 + i * Δx + 5) * Δx] from i=1 to n

As n approaches infinity, the width of each subinterval Δx approaches zero. Therefore, the limit is:

lim(n→∞) ∑[(x_i + 5) * Δx] from i=1 to n = ∫(x + 5) dx from 1 to 5

Final Answer

The final answer is:

∫(x + 5) dx from 1 to 5 = 30

Introduction

In our previous article, we explored how to express a definite integral as the limit of a Riemann sum. In this article, we will answer some common questions related to this topic.

Q: What is a Riemann sum?

Q: How do I calculate a Riemann sum?

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

Divide the interval into subintervals: Divide the interval [a, b] into n subintervals of equal width, denoted by Δx.
Choose a point in each subinterval: Choose a point in each subinterval, denoted by x_i.
Calculate the function value at each point: Calculate the function value at each point, denoted by f(x_i).
Calculate the area of each rectangle: Calculate the area of each rectangle by multiplying the function value at each point by the width of the subinterval, denoted by f(x_i) * Δx.
Sum the areas of the rectangles: Sum the areas of the rectangles to obtain the Riemann sum.

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

A Riemann sum is an approximation of the area under a curve, while a definite integral is the exact area under the curve. As the number of subintervals increases, the Riemann sum approaches the definite integral.

Q: How do I express a definite integral as a Riemann sum?

To express a definite integral as a Riemann sum, you need to follow these steps:

Divide the interval into subintervals: Divide the interval [a, b] into n subintervals of equal width, denoted by Δx.
Choose a point in each subinterval: Choose a point in each subinterval, denoted by x_i.
Calculate the function value at each point: Calculate the function value at each point, denoted by f(x_i).
Calculate the area of each rectangle: Calculate the area of each rectangle by multiplying the function value at each point by the width of the subinterval, denoted by f(x_i) * Δx.
Sum the areas of the rectangles: Sum the areas of the rectangles to obtain the Riemann sum.

Q: What is the formula for a Riemann sum?

The formula for a Riemann sum is:

∑[f(x_i) * Δx] from i=1 to n

where f(x_i) is the function value at each point, Δx is the width of each subinterval, and n is the number of subintervals.

Q: How do I evaluate the limit of a Riemann sum?

To evaluate the limit of a Riemann sum, you need to follow these steps:

Divide the interval into subintervals: Divide the interval [a, b] into n subintervals of equal width, denoted by Δx.
Choose a point in each subinterval: Choose a point in each subinterval, denoted by x_i.
Calculate the function value at each point: Calculate the function value at each point, denoted by f(x_i).
Calculate the area of each rectangle: Calculate the area of each rectangle by multiplying the function value at each point by the width of the subinterval, denoted by f(x_i) * Δx.
Sum the areas of the rectangles: Sum the areas of the rectangles to obtain the Riemann sum.
Take the limit as n approaches infinity: Take the limit of the Riemann sum as n approaches infinity to obtain the definite integral.

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

Q: How do I use Riemann sums in real-world applications?

Riemann sums are used in a variety of real-world applications, including:

Physics: Riemann sums are used to calculate the area under curves that represent physical quantities, such as velocity and acceleration.
Engineering: Riemann sums are used to calculate the area under curves that represent physical quantities, such as stress and strain.
Economics: Riemann sums are used to calculate the area under curves that represent economic quantities, such as supply and demand.

Conclusion

In this article, we have answered some common questions related to expressing definite integrals as Riemann sums. We have also discussed the relationship between Riemann sums and definite integrals, and provided examples of how Riemann sums are used in real-world applications.