Verification Using Step Functions

Oct 13, 2025 by pynkocean.com 72 views

Introduction

In a recent lecture on the definition of integrals in terms of step functions, my lecturer introduced an unusual function on the interval [0,1]. This function, known as a "weighting" function, has sparked my curiosity and led me to evaluate its integral. In this article, we will delve into the world of integration and explore the properties of this unusual function. We will also verify our solution using step functions and discuss the implications of this result.

The Unusual "Weighting" Function

The unusual "weighting" function is defined as:

f(x) = { 0, 0 ≤ x < 1/2 { 1, 1/2 ≤ x ≤ 1

This function is a step function, which is a piecewise function that consists of a finite number of constant values. The function has two distinct regions: one where the value is 0, and another where the value is 1. The function is discontinuous at x = 1/2, where the value jumps from 0 to 1.

Evaluating the Integral

To evaluate the integral of this unusual function, we can use the definition of the integral in terms of step functions. The integral of a function f(x) over the interval [a,b] is defined as:

∫[a,b] f(x) dx = lim(n→∞) ∑[i=1 to n] f(x_i) * (x_i - x_{i-1})

where x_i is a partition of the interval [a,b] into n subintervals, and f(x_i) is the value of the function at the point x_i.

Using this definition, we can evaluate the integral of the unusual "weighting" function over the interval [0,1]. We can partition the interval [0,1] into n subintervals, each of length 1/n. The points x_i will be the endpoints of these subintervals, and the value of the function at each point x_i will be either 0 or 1.

Calculating the Integral

Let's calculate the integral of the unusual "weighting" function over the interval [0,1]. We can use the definition of the integral in terms of step functions:

∫[0,1] f(x) dx = lim(n→∞) ∑[i=1 to n] f(x_i) * (x_i - x_{i-1})

We can simplify this expression by noting that the function f(x) is 0 for x < 1/2 and 1 for x ≥ 1/2. Therefore, the sum will only include terms where x_i ≥ 1/2.

For each term in the sum, we have:

f(x_i) * (x_i - x_{i-1}) = 1 * (x_i - x_{i-1})

Since x_i - x_{i-1} = 1/n, we have:

f(x_i) * (x_i - x_{i-1}) = 1/n

The sum will include n terms, each of which is 1/n. Therefore, the sum is:

∑[i=1 to n] f(x_i) * (x_i - x_{i-1}) = n * (1/n) = 1

Conclusion

We have evaluated the integral of the unusual "weighting" function over the interval [0,1] using the definition of the integral in terms of step functions. We found that the integral is equal to 1.

Verification Using Step Functions

To verify our result, we can use step functions to approximate the integral. A step function is a piecewise function that consists of a finite number of constant values. We can use a step function to approximate the unusual "weighting" function by dividing the interval [0,1] into n subintervals and assigning a constant value to each subinterval.

Let's use a step function to approximate the unusual "weighting" function. We can divide the interval [0,1] into n subintervals, each of length 1/n. We can assign a constant value to each subinterval, which will be either 0 or 1.

Calculating the Approximation

Let's calculate the approximation of the unusual "weighting" function using a step function. We can use the definition of the integral in terms of step functions:

∫[0,1] f(x) dx ≈ lim(n→∞) ∑[i=1 to n] f_i * (x_i - x_{i-1})

where f_i is the constant value assigned to the i-th subinterval, and x_i is the endpoint of the i-th subinterval.

We can simplify this expression by noting that the function f(x) is 0 for x < 1/2 and 1 for x ≥ 1/2. Therefore, the sum will only include terms where x_i ≥ 1/2.

For each term in the sum, we have:

f_i * (x_i - x_{i-1}) = 1 * (x_i - x_{i-1})

Since x_i - x_{i-1} = 1/n, we have:

f_i * (x_i - x_{i-1}) = 1/n

The sum will include n terms, each of which is 1/n. Therefore, the sum is:

∑[i=1 to n] f_i * (x_i - x_{i-1}) = n * (1/n) = 1

Conclusion

We have verified our result using a step function to approximate the unusual "weighting" function. We found that the integral is equal to 1.

Implications

The unusual "weighting" function has sparked my curiosity and led me to evaluate its integral. We have found that the integral is equal to 1, which has implications for the definition of the integral in terms of step functions.

The unusual "weighting" function is a step function, which is a piecewise function that consists of a finite number of constant values. The function has two distinct regions: one where the value is 0, and another where the value is 1. The function is discontinuous at x = 1/2, where the value jumps from 0 to 1.

The integral of the unusual "weighting" function over the interval [0,1] is equal to 1, which has implications for the definition of the integral in terms of step functions. The result suggests that the integral of a step function can be evaluated using the definition of the integral in terms of step functions.

Conclusion

Introduction

In our previous article, we evaluated the integral of an unusual "weighting" function over the interval [0,1] using the definition of the integral in terms of step functions. We found that the integral is equal to 1. In this article, we will answer some of the most frequently asked questions about the integral of the unusual "weighting" function.

Q: What is the unusual "weighting" function?

A: The unusual "weighting" function is a step function defined as:

f(x) = { 0, 0 ≤ x < 1/2 { 1, 1/2 ≤ x ≤ 1

This function is a piecewise function that consists of a finite number of constant values. The function has two distinct regions: one where the value is 0, and another where the value is 1. The function is discontinuous at x = 1/2, where the value jumps from 0 to 1.

Q: How did you evaluate the integral of the unusual "weighting" function?

A: We evaluated the integral of the unusual "weighting" function using the definition of the integral in terms of step functions. We divided the interval [0,1] into n subintervals, each of length 1/n. We then assigned a constant value to each subinterval, which was either 0 or 1. We used the definition of the integral in terms of step functions to evaluate the integral:

∫[0,1] f(x) dx = lim(n→∞) ∑[i=1 to n] f(x_i) * (x_i - x_{i-1})

where x_i is the endpoint of the i-th subinterval, and f(x_i) is the constant value assigned to the i-th subinterval.

Q: Why is the integral of the unusual "weighting" function equal to 1?

A: The integral of the unusual "weighting" function is equal to 1 because the function is 0 for x < 1/2 and 1 for x ≥ 1/2. Therefore, the sum will only include terms where x_i ≥ 1/2. For each term in the sum, we have:

f(x_i) * (x_i - x_{i-1}) = 1 * (x_i - x_{i-1})

Since x_i - x_{i-1} = 1/n, we have:

f(x_i) * (x_i - x_{i-1}) = 1/n

The sum will include n terms, each of which is 1/n. Therefore, the sum is:

∑[i=1 to n] f(x_i) * (x_i - x_{i-1}) = n * (1/n) = 1

Q: Can you verify the result using a step function?

A: Yes, we can verify the result using a step function to approximate the unusual "weighting" function. We can divide the interval [0,1] into n subintervals, each of length 1/n. We can assign a constant value to each subinterval, which will be either 0 or 1. We can then use the definition of the integral in terms of step functions to evaluate the integral:

∫[0,1] f(x) dx ≈ lim(n→∞) ∑[i=1 to n] f_i * (x_i - x_{i-1})

where f_i is the constant value assigned to the i-th subinterval, and x_i is the endpoint of the i-th subinterval.

We can simplify this expression by noting that the function f(x) is 0 for x < 1/2 and 1 for x ≥ 1/2. Therefore, the sum will only include terms where x_i ≥ 1/2.

For each term in the sum, we have:

f_i * (x_i - x_{i-1}) = 1 * (x_i - x_{i-1})

Since x_i - x_{i-1} = 1/n, we have:

f_i * (x_i - x_{i-1}) = 1/n

The sum will include n terms, each of which is 1/n. Therefore, the sum is:

∑[i=1 to n] f_i * (x_i - x_{i-1}) = n * (1/n) = 1

Q: What are the implications of the result?

A: The result has implications for the definition of the integral in terms of step functions. The unusual "weighting" function is a step function, which is a piecewise function that consists of a finite number of constant values. The function has two distinct regions: one where the value is 0, and another where the value is 1. The function is discontinuous at x = 1/2, where the value jumps from 0 to 1.

The integral of the unusual "weighting" function over the interval [0,1] is equal to 1, which suggests that the integral of a step function can be evaluated using the definition of the integral in terms of step functions.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about the integral of the unusual "weighting" function. We have evaluated the integral using the definition of the integral in terms of step functions and verified the result using a step function to approximate the unusual "weighting" function. The result has implications for the definition of the integral in terms of step functions.