Problem 84 - Calculus of a Single Variable

Question

True or False? In Exercises \(83-86\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \begin{array}{l}{\text { If }} \\ {\int_{a}^{b}[f(x)-g(x)] d x=A} \\ {\text { then }} \\ {\int_{a}^{b}[g(x)-f(x)] d x=-A}\end{array} $$

Step-by-Step Solution

Answer

The statement 'If \( \int_{a}^{b} [f(x) - g(x)] dx = A \), then \( \int_{a}^{b} [g(x) - f(x)] dx = -A \)' is True.

Step 1: Understanding the given statement

Given \( \int_{a}^{b}[f(x)-g(x)] d x = A \), it needs to be understood if \( \int_{a}^{b}[g(x)-f(x)] d x = -A \), essentially if switching the places of the subtrahend and minuend in the integral affects the integral's sign.

Step 2: Identifying the property of integrals

The additive property of integrals indicates that \( \int_{a}^{b} [f(x) \pm g(x)] dx = \int_{a}^{b} f(x) dx \pm \int_{a}^{b} g(x) dx \). Thus, the definite integral of the difference of two functions is equal to the difference of the definite integrals of the functions.

Step 3: Applying the property

Applying this property to the problem at hand, we know that \( \int_{a}^{b} [f(x) - g(x)] dx = A \). So, reversing the subtrahend and minuend will change the sign of the integral. Hence, \( \int_{a}^{b} [g(x) - f(x)] dx = -A \)

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that calculates the accumulation of quantities. It is denoted as \( \int_{a}^{b} f(x) \, dx \), where the limits \( a \) and \( b \) define the interval over which the function \( f(x) \) is integrated. The resulting value is a numerical representation of the area under the curve of the function from \( a \) to \( b \). This concept is highly useful not only for calculating areas but also for finding out distances, probabilities, and more in different domains.
A few points to always remember:

• The definite integral gives the net area, considering areas below the x-axis as negative.
• A positive result indicates a net gain of area above the x-axis.
• Swapping the limits of integration, like \( a \) and \( b \), reverses the sign of the integral result.

Understanding definite integrals is crucial for solving problems that involve accumulation and change.

Integral Properties

Integral properties are rules that help simplify and manipulate integrals for easier calculation. These properties ensure that you can break down complex expressions into simpler parts. Here are some key properties:

• Linearity Property: \( \int_{a}^{b} [cf(x) + dg(x)] \, dx = c \int_{a}^{b} f(x) \, dx + d \int_{a}^{b} g(x) \, dx \) for constants \( c \) and \( d \).
• Additive Property: The integral of a sum or difference of functions is equivalent to the sum or difference of their integrals: \( \int_{a}^{b} [f(x) \pm g(x)] \, dx = \int_{a}^{b} f(x) \, dx \pm \int_{a}^{b} g(x) \, dx \).
• Reversal of Limits: Flipping the integration limits changes the sign: \( \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \).

These properties are pivotal in calculus, especially when dealing with the algebraic manipulation of integrals, making complex problems more approachable.

Function Difference

In calculus, the difference between two functions, such as \( f(x) - g(x) \), plays an important role in various calculations and transformations. The integral of such a difference, \( \int_{a}^{b} [f(x) - g(x)] \, dx \), can be handled by using the additive property of integrals, yielding \( \int_{a}^{b} f(x) \, dx - \int_{a}^{b} g(x) \, dx \).
But what happens if we switch the order? By switching the function positions, \( [g(x) - f(x)] \), the sign of the integral is altered, as seen in the statement \( \int_{a}^{b}[g(x)-f(x)] \, dx = -A \). This occurs because:

• The function difference is reversed, leading to negation.
• This is similar to saying \( -(f(x) - g(x)) \), which results in \( g(x) - f(x) \).

Understanding this concept helps to appreciate how reversals in subtraction influence overall outcomes.

Mathematical Proof

Proof is the backbone of mathematical validity, serving as a logical demonstration that a statement or theorem holds true universally.
In the case of integral differences, the logical reasoning follows directly from proven integral properties. To prove the original statement, consider:

• Given: \( \int_{a}^{b} [f(x) - g(x)] \, dx = A \).
• Act of reversing: \( \int_{a}^{b} [g(x) - f(x)] \, dx = -A \), applying the integral reversal property.
• Verification: This reversal follows from the initial definition of difference, where changing the order results in negation.

The solid foundation of mathematical proof ensures that such integral operations and transformations are reliable and consistent, providing clarity and certainty in problem-solving.