Riemann Integrability Of F(x) = 2 - 3x On [1, 3] A Detailed Proof
In this comprehensive exploration, we delve into the Riemann integrability of the function f(x) = 2 - 3x over the closed interval [1, 3]. This article aims to provide a detailed, step-by-step analysis, ensuring a clear understanding of the concepts involved. We will explore the properties of the function, the implications of its decreasing nature, and the application of the Riemann integrability criteria. Our goal is to present a rigorous yet accessible explanation suitable for students and enthusiasts of mathematical analysis. Understanding Riemann integrability is crucial for various applications in calculus and real analysis, making this a fundamental topic in mathematical education. The concepts discussed here are essential for anyone looking to deepen their understanding of integration beyond the basic computations and into the theoretical underpinnings.
The Riemann integral is a cornerstone of calculus, providing a rigorous definition for the integral of a function. It's essential to grasp the nuances of Riemann integrability to understand the scope and limitations of integration. In this discussion, we will meticulously examine how the properties of f(x) = 2 - 3x, particularly its decreasing behavior over the interval [1, 3], influence its Riemann integrability. We'll also connect these concepts to broader theorems and principles in real analysis, illustrating the significance of this analysis within the larger framework of mathematical theory. This article serves not only as a solution to the specific problem but also as a tutorial on how to approach similar problems involving Riemann integrability. By the end of this discussion, you should have a solid grasp of the conditions required for a function to be Riemann integrable and how to verify these conditions in practical scenarios.
Before diving into the specifics of our function, let's lay the groundwork by defining Riemann integrability. A function f(x) is said to be Riemann integrable over an interval [a, b] if the Riemann integral of f(x) from a to b exists. To formally define this, we need to introduce the concepts of partitions, Riemann sums, and upper and lower sums. A partition P of the interval [a, b] is a finite set of points x₀, x₁, ..., xₙ}* such that a = x₀ < x₁ < ... < xₙ = b. The norm of the partition, denoted as ||P||, is the length of the largest subinterval [xᵢ₋₁, xᵢ]. For each subinterval [xᵢ₋₁, xᵢ], we can define *mᵢ = inf{f(x) and Mᵢ = supf(x) . The lower Riemann sum L(f, P) is then defined as ∑ᵢ₌₁ⁿ mᵢ(xᵢ - xᵢ₋₁), and the upper Riemann sum U(f, P) is defined as ∑ᵢ₌₁ⁿ Mᵢ(xᵢ - xᵢ₋₁).
The lower Riemann integral is the supremum of all lower sums, and the upper Riemann integral is the infimum of all upper sums. A function f(x) is Riemann integrable if and only if the lower Riemann integral equals the upper Riemann integral. In simpler terms, this means that as we refine our partitions (i.e., make the subintervals smaller), the upper and lower sums converge to the same value. This value is the Riemann integral of f(x) over [a, b]. This definition is crucial for understanding the theoretical underpinnings of integration. The convergence of Riemann sums to a single value is what allows us to define the area under the curve rigorously. The concept of upper and lower sums provides a way to bound the integral from above and below, and their convergence ensures that the integral is well-defined. Furthermore, this definition allows us to establish criteria for Riemann integrability, such as the condition that continuous functions and monotone functions are Riemann integrable.
Now, let's focus on the function f(x) = 2 - 3x over the interval [1, 3]. First and foremost, observe that f(x) is a linear function. Linear functions are continuous everywhere, and continuity is a sufficient condition for Riemann integrability. However, the problem statement also highlights that f(x) is decreasing on [1, 3], which is another crucial property we can leverage. To confirm that f(x) is indeed decreasing, we can compute its derivative, f'(x) = -3. Since the derivative is negative for all x, the function is strictly decreasing over its entire domain, including the interval [1, 3]. This decreasing nature has significant implications for our analysis of Riemann integrability because it simplifies the process of determining the upper and lower sums.
Given that f(x) = 2 - 3x is decreasing, for any subinterval [xᵢ₋₁, xᵢ] within our partition, the maximum value Mᵢ of f(x) will occur at the left endpoint xᵢ₋₁, and the minimum value mᵢ will occur at the right endpoint xᵢ. This makes it straightforward to calculate the upper and lower Riemann sums. Specifically, Mᵢ = f(xᵢ₋₁) and mᵢ = f(xᵢ). This characteristic of decreasing functions greatly simplifies the computation of Riemann sums and the subsequent analysis of integrability. The decreasing nature also ensures that the difference between the upper and lower sums can be controlled by refining the partition, which is a key step in proving Riemann integrability. Furthermore, understanding this behavior helps in visualizing the Riemann sums as approximations of the area under the curve, with the upper sum overestimating the area and the lower sum underestimating it. The convergence of these sums as the partition becomes finer confirms the Riemann integrability of the function.
To formally verify that f(x) = 2 - 3x is Riemann integrable over [1, 3], we can use the Riemann integrability criterion. This criterion states that a function f is Riemann integrable on [a, b] if and only if for every ε > 0, there exists a partition P of [a, b] such that U(f, P) - L(f, P) < ε. In other words, we need to show that we can make the difference between the upper and lower sums arbitrarily small by choosing a suitable partition. For our function f(x) = 2 - 3x, let's consider a uniform partition P of [1, 3] into n equal subintervals. The length of each subinterval is Δx = (3 - 1)/n = 2/n. The partition points are xᵢ = 1 + i(2/n) for i = 0, 1, ..., n.
Since f(x) is decreasing, we have Mᵢ = f(xᵢ₋₁) = 2 - 3(1 + (i - 1)(2/n)) and mᵢ = f(xᵢ) = 2 - 3(1 + i(2/n)). The upper Riemann sum is U(f, P) = ∑ᵢ₌₁ⁿ MᵢΔx = ∑ᵢ₌₁ⁿ 2 - 3(1 + (i - 1)(2/n)), and the lower Riemann sum is L(f, P) = ∑ᵢ₌₁ⁿ mᵢΔx = ∑ᵢ₌₁ⁿ 2 - 3(1 + i(2/n)). Now we compute the difference U(f, P) - L(f, P):
U(f, P) - L(f, P) = ∑ᵢ₌₁ⁿ f(xᵢ₋₁) - f(xᵢ) = ∑ᵢ₌₁ⁿ 2 - 3(1 + (i - 1)(2/n)) - (2 - 3(1 + i(2/n)))
Simplifying the expression inside the summation, we get:
U(f, P) - L(f, P) = ∑ᵢ₌₁ⁿ 2 - 3 - 3(i - 1)(2/n) - 2 + 3 + 3i(2/n) = ∑ᵢ₌₁ⁿ 3(2/n) = ∑ᵢ₌₁ⁿ (12/n²)
Since the term inside the summation is constant with respect to i, the sum is simply n(12/n²) = 12/n*. Now, for any ε > 0, we want to find an n such that U(f, P) - L(f, P) = 12/n < ε. Solving for n, we get n > 12/ε. Thus, for any given ε > 0, we can choose n to be an integer greater than 12/ε, and the difference between the upper and lower sums will be less than ε. This satisfies the Riemann integrability criterion.
To further clarify the application of the Riemann integrability criterion, let's provide a more detailed calculation of the difference between the upper and lower sums. We have established that:
U(f, P) - L(f, P) = ∑ᵢ₌₁ⁿ f(xᵢ₋₁) - f(xᵢ)
Given f(x) = 2 - 3x and xᵢ = 1 + i(2/n), we have:
f(xᵢ₋₁) = 2 - 3(1 + (i - 1)(2/n)) = 2 - 3 - 6(i - 1)/n = -1 - 6(i - 1)/n f(xᵢ) = 2 - 3(1 + i(2/n)) = 2 - 3 - 6i/n = -1 - 6i/n
So,
f(xᵢ₋₁) - f(xᵢ) = (-1 - 6(i - 1)/n) - (-1 - 6i/n) = -1 - 6i/n + 6/n + 1 + 6i/n = 6/n
Therefore,
U(f, P) - L(f, P) = ∑ᵢ₌₁ⁿ (6/n)(2/n) = ∑ᵢ₌₁ⁿ (12/n²) = n(12/n²) = 12/n
As we showed before, for any ε > 0, we can choose n > 12/ε to ensure that U(f, P) - L(f, P) < ε. This detailed calculation reinforces our conclusion that f(x) = 2 - 3x is Riemann integrable over [1, 3]. The explicit computation of the difference between the upper and lower sums demonstrates how the decreasing nature of the function allows us to control this difference by increasing the number of subintervals in the partition. This approach is a standard technique in real analysis for proving the Riemann integrability of monotone functions.
In conclusion, we have rigorously demonstrated that the function f(x) = 2 - 3x is Riemann integrable over the interval [1, 3]. We first established the definition of Riemann integrability and the criterion based on the difference between upper and lower Riemann sums. We then analyzed the function f(x), noting its continuity and decreasing nature. This decreasing property was crucial in simplifying the calculation of the upper and lower sums. By constructing a uniform partition and explicitly computing the difference between the upper and lower sums, we showed that this difference can be made arbitrarily small by choosing a sufficiently fine partition. This satisfies the Riemann integrability criterion, thus proving that f(x) is indeed Riemann integrable.
This analysis highlights the importance of understanding the properties of functions when determining their Riemann integrability. The combination of continuity and monotonicity provides a straightforward path to verifying integrability using the Riemann integrability criterion. This detailed exploration provides a solid foundation for understanding more advanced topics in real analysis and calculus. Furthermore, the techniques demonstrated here can be applied to a wide range of functions and intervals, making this a valuable skill for anyone studying mathematical analysis. The concept of Riemann integrability is fundamental to the theory of integration, and a thorough understanding of this concept is essential for further studies in mathematics and related fields.
