Question

(a) Suppose that \(f\) is integrable on a rectangle \(R\) and \(P=\left\\{R_{1}, R_{2}, \ldots, R_{k}\right\\}\) is a partition of \(R\). Show that $$ \int_{R} f(\mathbf{X}) d \mathbf{X}=\sum_{j=1}^{k} \int_{R_{j}} f(\mathbf{X}) d \mathbf{X} $$ (b) Use (a) to show that if \(f\) is continuous on \(R\) and \(P\) is a partition of \(R,\) then there is a Riemann sum of \(f\) over \(P\) that equals \(\int_{R} f(\mathbf{X}) d \mathbf{X}\)

Short answer

In the given problem, we proved that if a function is integrable on a rectangle and the rectangle is partitioned into smaller rectangles, then the integral of the function over the larger rectangle is equal to the sum of integrals over each smaller rectangle. Then, we showed that for a continuous function on a rectangle, there exists a Riemann sum over the partition of the rectangle that is equal to the integral of the function over the rectangle.

Step-by-step solution

Part (a) - Proof of the equality of integrals

Let \(f\) be integrable on a rectangle \(R\). If \(R\) is partitioned into smaller rectangles, which we denote as \(R_1, R_2, ..., R_k\), then we can write \(R = \bigcup_{j=1}^k R_j\). To calculate the integral of \(f\) over \(R\), we will consider the Riemann sum of \(f\) over the partition \(P=\{R_1, R_2, ..., R_k\}\) which is given by: $$ \sum_{j=1}^{k} M_j \Delta R_j $$ Here, \(M_j = \sup_{\mathbf{X} \in R_j} f(\mathbf{X})\) and \(\Delta R_j\) represents the volume of the j-th subrectangle \(R_j\). Since \(f\) is integrable, the Riemann sum converges to the integral of \(f\) over \(R\) as the mesh size of the partition approaches zero, that is: $$ \int_{R} f(\mathbf{X}) d \mathbf{X} = \sum_{j=1}^{k} \lim_{\|\Delta R_j\| \rightarrow 0} M_j \Delta R_j $$ Now, for each j-th subrectangle \(R_j\) with \(M_j\) and \(\Delta R_j\), we can see that as the mesh size of the partition approaches zero, the Riemann sum for \(f\) over \(R_j\) is converging to the integral of \(f\) over \(R_j\). Therefore, we can write: $$ \int_{R_j} f(\mathbf{X}) d \mathbf{X} = \lim_{\|\Delta R_j\| \rightarrow 0} M_j \Delta R_j $$ Now, substituting this back into the original equation, we get: $$ \int_{R} f(\mathbf{X}) d \mathbf{X} = \sum_{j=1}^{k} \int_{R_j} f(\mathbf{X}) d \mathbf{X} $$ Hence, the equality is proven for part (a).

Part (b) - Finding the Riemann sum of continuous function

Let \(f\) be a continuous function on a rectangle \(R\), and let \(P=\{R_1, R_2, ..., R_k\}\) be a partition of R. Using the result from part (a), we have: $$ \int_{R} f(\mathbf{X}) d \mathbf{X} = \sum_{j=1}^{k} \int_{R_j} f(\mathbf{X}) d \mathbf{X} $$ Since \(f\) is continuous, we can apply the intermediate value theorem which states that, for each subrectangle \(R_j\), there exists a point \(\mathbf{X}_j \in R_j\) such that: $$ \int_{R_j} f(\mathbf{X}) d \mathbf{X} = f(\mathbf{X}_j) \Delta R_j $$ Substituting this back into the previous equation, we get: $$ \int_{R} f(\mathbf{X}) d \mathbf{X} = \sum_{j=1}^{k} f(\mathbf{X}_j) \Delta R_j $$ This equation represents a Riemann sum of \(f\) over the partition \(P\), with the function evaluated at the points \(\mathbf{X}_j\) in each subrectangle. Thus, we have shown that there is a Riemann sum of a continuous function \(f\) over a partition \(P\) that equals the integral of \(f\) over \(R\).

Key concepts

Real Analysis

Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions. Understanding the concept of a Riemann integral is fundamental in real analysis, as it provides the groundwork for defining and working with integrals in a precise manner. An integral, in simple terms, is a mathematical representation of the total accumulation of quantities, such as area under a curve.

In the context of our exercise, real analysis comes into play when we examine the properties of integrable functions. The exercise kicks off by considering a function that is integrable over a certain region, which leads to exploring how the integral over a larger region can be broken down into integrals over smaller, constituent regions. This breaking down process involves using partitions and understanding the behavior of continuous functions, two key aspects of real analysis.

Partition of a Rectangle

A partition of a rectangle in real analysis refers to dividing a rectangular region into a finite number of non-overlapping smaller rectangles. This concept is crucial when working with Riemann integrals because it allows for the approximation of an integral by summing the areas of rectangles that fit inside the region under a curve.

In the context of the exercise, a partition \(P\) divides a large rectangle \(R\) into smaller rectangles \(R_1, R_2, ..., R_k\). Each of these smaller rectangles can be used to approximate the integral over \(R\) individually, with their respective areas potentially providing a better representation of the total integral as they become finer. The overall purpose of partitioning is to make the problem more manageable and to provide a clear path towards solving for the integral using Riemann sums.

Continuity in Real Analysis

Continuity of a function is a central concept in real analysis. A function is continuous at a point if small changes in the input result in small changes in the output. In more formal terms, given any small positive number \(\epsilon\), there exists another small positive number \(\delta\) such that whenever the difference between the input and the point is less than \(\delta\), the difference between the corresponding outputs is less than \(\epsilon\).

Continuity is highly relevant for our exercise when working with integrals. In the second part of the problem, we use the fact that if a function \(f\) is continuous over a rectangle \(R\), then we can apply the intermediate value theorem. This theorem guarantees the existence of a specific output value for every value in the range of \(f\), for a given sub-rectangle from our partition. It allows us to find specific points within each sub-rectangle that contribute to a Riemann sum that precisely equals the integral over \(R\).

Riemann Sum

The Riemann sum is a method to approximate the integral of a function over an interval. It's calculated by dividing the interval into smaller subintervals, finding the value of the function at specific points within these subintervals, and then summing up the products of these values and the lengths of the subintervals.

The exercise demonstrates the relationship between Riemann sums and Riemann integrals. When the function \(f\) is continuous and the partition \(P\) is given, we can select points \(\mathbf{X}_j\) within each sub-rectangle so that the sum of the products of \(f(\mathbf{X}_j)\) and the volume of \(R_j\) gives us a Riemann sum that equals the integral over the entire rectangle \(R\). This establishes a direct connection between the partitioned areas and the original integral, defining a clear approach to calculating the integral through discrete sums of partitioned rectangles. The concept of the Riemann sum not only simplifies the process of integration but also provides an intuitive understanding of the accumulation of quantities.