Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove the Cauchy-Schwarz Inequality for Integrals: \(\left[\int_{a}^{b} f(x) g(x) d x\right]^{2} \leq \int_{a}^{b} f^{2}(x) d x \int_{a}^{b} g^{2}(x) d x\) Hint: Consider the double integral of $$ F(x, y)=[f(x) g(y)-f(y) g(x)]^{2} $$ over the rectangle \(R=\\{(x, y): a \leq x \leq b, a \leq y \leq b\\}\).

Step-by-Step Solution

Answer

The inequality is proven using the non-negative property of squared expressions and double integrals.

Step 1: Rewrite the Expression

Start by considering the function \( F(x, y) = [f(x) g(y) - f(y) g(x)]^2 \). The goal is to explore the integral of \( F(x,y) \) over the rectangle \( R = \{(x, y): a \leq x \leq b, a \leq y \leq b\} \). Thus, we have the double integral \( \int_{a}^{b} \int_{a}^{b} [f(x) g(y) - f(y) g(x)]^2 \, dy \, dx \).

Step 2: Expand the Square

Expand \( [f(x) g(y) - f(y) g(x)]^2 \) to get \( f^2(x)g^2(y) - 2f(x)g(x)f(y)g(y) + f^2(y)g^2(x) \). This will simplify our calculations when we integrate over \( R \).

Step 3: Integrate Term by Term

For the first term, integrate \( \int_{a}^{b} \int_{a}^{b} f^2(x) g^2(y) \, dy \, dx = \left( \int_{a}^{b} f^2(x) \, dx \right) \left( \int_{a}^{b} g^2(y) \, dy \right) \). Use Fubini's Theorem for this independent integration.

Step 4: Integrate the Cross Term

For the cross term, which is \(-2 \int_{a}^{b} \int_{a}^{b} f(x) g(x) f(y) g(y) \, dy \, dx \), the function factors into separate integrals: \(- 2 \int_{a}^{b} f(x) g(x) \, dx \cdot \int_{a}^{b} f(y) g(y) \, dy \).

Step 5: Integrate the Second Square Term

For the second square term, integrate \( \int_{a}^{b} \int_{a}^{b} f^2(y) g^2(x) \, dy \, dx \). Use similar logic from step 3, giving \( \left( \int_{a}^{b} f^2(y) \, dy \right) \left( \int_{a}^{b} g^2(x) \, dx \right) \).

Step 6: Combine All Terms

Sum the results from Steps 3, 4, and 5. You will have: \[ \int_{a}^{b} \int_{a}^{b} [f(x) g(y) - f(y) g(x)]^2 \, dy \, dx = \int_{a}^{b} f^2(x) \, dx \int_{a}^{b} g^2(y) \, dy + \int_{a}^{b} f^2(y) \, dy \int_{a}^{b} g^2(x) \, dx - 2 \left( \int_{a}^{b} f(x) g(x) \, dx \right)^2 \]

Step 7: Conclude the Inequality

Since \( \int_{a}^{b} \int_{a}^{b} [f(x) g(y) - f(y) g(x)]^2 \, dy \, dx \geq 0 \) (as it is the integral of a square), it follows that: \[ \left( \int_{a}^{b} f(x) g(x) \, dx \right)^2 \leq \int_{a}^{b} f^2(x) \, dx \int_{a}^{b} g^2(y) \, dy \]. This completes the proof of the Cauchy-Schwarz inequality for integrals.

Key Concepts

Integrals

An integral is a mathematical concept used to calculate the accumulation of quantities. Imagine it as adding up a series of tiny pieces to find the total. For example, you might use an integral to calculate the area under a curve or the total distance traveled by an object moving at a variable speed.
Think of it as a method to combine or accumulate several small parts into a whole. When you need to integrate a function, you are essentially summing up these tiny slices over a certain interval.

• Definite integrals, such as \( \int_{a}^{b} f(x) \, dx \), provide a number that represents the area under the function from point \( a \) to point \( b \).
• Indefinite integrals, on the other hand, give you a new function whose derivative matches the original function.

Understanding integrals deeply is crucial for grasping advanced mathematical concepts such as double integrals, which expand this idea into more dimensions.

Double Integrals

Building on the concept of single integrals, double integrals allow us to integrate functions over a two-dimensional area. It is like adding up the volume under a surface in three-dimensional space.

For a function \( z = f(x, y) \) over a region \( R \), the double integral \[ \int \int_{R} f(x, y) \, dA \] represents the volume under the surface defined by the function, across the specified region.

• In terms of calculations, you first integrate with respect to one variable while treating the other as constant, and then proceed to the second integration.
• Double integrals are widely used in physics and engineering to calculate quantities like mass, area, and volume of physically varied shapes.

By practicing with double integrals, you can better visualize how mathematical concepts apply to the real world, especially as they relate to two-dimensional regions.

Fubini's Theorem

Fubini's Theorem is an invaluable tool when dealing with double integrals. It states that under certain conditions, the order of integration can be changed without affecting the result of the integral. This is especially useful when one order of integration is much easier to compute than the other.

To apply Fubini's Theorem, the function must be continuous on the closed region you are integrating over. It allows us to express a double integral as two consecutive single integrals:\[ \int \int_{R} f(x, y) \, dA = \int_{a}^{b} \left( \int_{c}^{d} f(x, y) \, dy \right) \, dx = \int_{c}^{d} \left( \int_{a}^{b} f(x, y) \, dx \right) \, dy \]

• This change can simplify the calculation and potentially make the problem more tractable.
• It is typically used when evaluating integrals over rectangular regions or other simple shapes.

Mastering Fubini's Theorem helps in tackling complex integration problems by finding the most efficient path to the solution.

Mathematical Proofs

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. Proving a theorem ensures that it holds under all possible circumstances covered by its assumptions. Mathematical proofs can involve different methods depending on the nature of the proposition.

• The idea is not only to verify if something is true, but also to understand why it is true.
• Direct proofs involve straightforward argumentation by logical steps. Indirect proofs, such as proof by contradiction, assume the opposite of what you want to prove and show that it leads to a contradiction.
• Constructive proofs provide explicit examples demonstrating the truth, while non-constructive proofs show existence without necessarily providing a witness.

To prove inequalities like the Cauchy-Schwarz Inequality in the context of integrals, one often needs to employ techniques like substitution or recognizing patterns. Understanding proofs is not only central to higher mathematics but also enhances critical thinking skills.