Problem 29 - University Calculus: Early Transcendentals

Question

Suppose \(f(x, y)\) is continuous over a region \(R\) in the plane and that the area \(A(R)\) of the region is defined. If there are constants \(m\) and \(M\) such that \(m \leq f(x, y) \leq M\) for all \((x, y) \in R,\) prove that $$ m A(R) \leq \iint_{k} f(x, y) d A \leq M A(R) $$

Step-by-Step Solution

Answer

The inequality is proved using the bounds \(m\) and \(M\) for the continuous function \(f(x, y)\) over region \(R\).

Step 1: Understand the Problem

We need to prove that if a function \(f(x, y)\) is continuous over a region \(R\) and is bounded by constants \(m\) and \(M\), then \(m A(R) \leq \iint_{R} f(x, y) \, dA \leq M A(R)\), where \(A(R)\) is the area of the region \(R\).

Step 2: Express the Double Integral

The double integral \(\iint_{R} f(x, y) \, dA\) represents the sum of all values of \(f(x, y)\) over the area of \(R\). This can be viewed as a weighted sum of function values scaled by infinitesimal area elements \(dA\).

Step 3: Relate Function Values to Bounds

Since \(m \leq f(x, y) \leq M\) for all \((x,y) \in R\), this implies that the value of each \(f(x, y)\) is at least \(m\) and at most \(M\) for every point in the region. Therefore, \[ m \cdot dA \leq f(x, y) \, dA \leq M \cdot dA. \]

Step 4: Integrate Over the Region

Integrating the inequalities in Step 3 over the entire region \(R\) gives: \[ \iint_{R} m \, dA \leq \iint_{R} f(x, y) \, dA \leq \iint_{R} M \, dA. \]

Step 5: Compute Area-Related Integrals

The integrals \(\iint_{R} m \, dA\) and \(\iint_{R} M \, dA\) simplify to \(m A(R)\) and \(M A(R)\), respectively, because \(m\) and \(M\) are constants. So, \[ m A(R) \leq \iint_{R} f(x, y) \, dA \leq M A(R). \]

Step 6: Conclude the Proof

We have shown that the inequality \(m A(R) \leq \iint_{R} f(x, y) \, dA \leq M A(R)\) holds, proving the statement as required.

Key Concepts

Continuous Functions

Continuous functions are essential in calculus because they ensure smoothness and predictability of behaviors within a region. When a function is continuous over a region, it means there are no sudden jumps or breaks in its graph. Here are some key points about continuous functions:

• A function is continuous at a point if the limit as it approaches that point equals the function's value at the point.
• A function of two variables, like our function \(f(x,y)\), is continuous over a region \(R\) if its partial derivatives exist and are continuous in \(R\).
• Continuous functions are crucial for applying theorems like the Fundamental Theorem of Calculus and proving inequalities involving double integrals.

When working with a continuous function, like in our problem, it allows us to use double integration and apply bounds across an entire region seamlessly.

Area of a Region

The concept of the "area of a region" is central to understanding how double integrals work. When we talk about the area of a region \(R\), we refer to the entire space in the plane that the region covers.Some important details include:

• The area of a simple, bounded region \(R\) in the plane can often be calculated as a straightforward integral \(\iint_R \, dA\).
• In the context of double integrals, the area element \(dA\) represents an infinitesimally small piece of this region across which our function is evaluated.
• Knowing the area \(A(R)\) allows us to scale the values of \(f(x, y)\) when demonstrating inequalities, as we did when showing \(m A(R) \leq \iint_R f(x, y) \, dA \leq M A(R)\).

Understanding the area helps us to integrate functions over regions effectively, making it possible to derive useful mathematical insights.

Inequalities in Calculus

Inequalities play a significant role in calculus, particularly in bounding the integrals and understanding function behavior. Let's break down how inequalities help us work with integrals:In our exercise, we dealt with:

• Finding constants \(m\) and \(M\) such that \(m \leq f(x, y) \leq M\) for all \((x, y) \in R\), ensuring the function is bounded within the region.
• The inequality \(m \cdot dA \leq f(x, y) \, dA \leq M \cdot dA\) implies, for each of the small elements \(dA\), the corresponding values of the function are also bounded by these constants.
• By integrating these inequalities over the entire region \(R\), we extend this boundary condition to include the whole area. This solidifies our understanding that the integral of the function \(f(x, y)\) is also proportionally bounded.

Using inequalities this way helps us define limits of integrals. It is a practical approach to problem-solving in calculus and ensures that calculations account for the maximum and minimum values a function can take over certain areas.