Problem 20 - The Calculus with Analytic Geometry

Question

If \(f\) is continuous on \([a, b]\), prove that $$ \begin{gathered} \left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x \\ (\text { HINT: }-|f(x)| \leq f(x) \leq|f(x)| .) \end{gathered} $$

Step-by-Step Solution

Answer

Applying the hint and integrating, we find that \ \left| \int_{a}^{b} f(x) \ dx \right| \leq \int_{a}^{b} |f(x)| \ dx \.

Step 1: Understand the problem

You need to prove that the absolute value of the integral of a continuous function over \([a, b]\) is less than or equal to the integral of the absolute value of the function over \([a, b]\). The hint is given in the form of an inequality about the function.

Step 2: Apply the hint

Given the hint: \(-|f(x)| \leq f(x) \leq |f(x)|\), we can integrate both sides of this inequality over \([a, b]\).

Step 3: Integrate the inequality

Integrate \(-|f(x)| \leq f(x) \leq |f(x)|\) from \a\ to \b\. \[ \int_a^b -|f(x)| \, dx \leq \int_a^b f(x) \, dx \leq \int_a^b |f(x)| \, dx \]

Step 4: Use the property of integrals

Since \ \int_a^b -|f(x)| \, dx\ is the negative of \ \int_a^b |f(x)| \, dx\, we have: \[-\int_a^b |f(x)| \, dx \leq \int_a^b f(x) \, dx \leq \int_a^b |f(x)| \, dx\]

Step 5: Analyze the inequalities

From the inequality \[-\int_a^b |f(x)| \, dx \leq \int_a^b f(x) \, dx\], we get that the integral \ \int_a^b f(x) \, dx \ is bounded by the positive value of \ \int_a^b |f(x)| \, dx \ on both sides.

Step 6: Conclude with absolute values

Therefore, taking absolute values gives: \[|\int_a^b f(x) \, dx | \leq \int_a^b |f(x)| \, dx \] This completes the proof.

Key Concepts

Continuous Functions

In mathematics, a function is considered continuous if, intuitively speaking, its graph can be drawn without lifting the pen from the paper. More formally, a function \(f\) is continuous on an interval \([a, b]\) if for every point \(c\) within this interval, the limit of \(f(x)\) as \(x\) approaches \(c\) equals \(f(c)\). This ensures there are no jumps, breaks, or holes in the function on \([a, b]\).
Continuous functions are vital in calculus because they ensure that integrals and derivatives behave in predictable ways.
The fundamental theorem of calculus, for instance, relies on the continuity of functions to link differentiation and integration.
In the exercise, the continuity of \(f\) on \([a, b]\) guarantees that we can apply integration and absolute value properties correctly. By knowing the function is continuous, we ensure that the limits and values involved are well-defined and can be manipulated using standard calculus principles.

Integration Inequalities

Integration inequalities provide essential tools for comparing the integrals of functions. These inequalities allow us to make statements about the size of an integral without computing it directly.
In the exercise, we used the inequality: -\(|f(x)|\) ≤ \(f(x)\) ≤ \(|f(x)|\) to establish boundaries for our original integral.
By integrating this inequality over the interval \([a, b]\), we applied integral properties to show: ∫_a^b -|f(x)| \, dx ≤ ∫_a^b f(x) \, dx ≤ ∫_a^b |f(x)| \, dx
This demonstrates how the integral of a function is bounded by the integral of its absolute value. Such integration inequalities are fundamental for establishing results in analysis, probability theory, and other areas.

Absolute Value Properties

The absolute value function, denoted as \(|x|\), measures the distance of a number \(x\) from zero on the real number line, ignoring its sign. Basic properties include:

• \(|x| \, ≥ \, 0\) for all real numbers \(x\)
• \(|x| = x\) if \(x \, ≥ \, 0\)
• \(|x| = -x\) if \(x \, < \,0\)
• \(|xy| = |x||y|\)
• \(|x+y| \, ≤ \,|x| + |y|\) (Triangle Inequality)

In our exercise, the absolute value of the integral showed how an integral's magnitude relates to the function's integral. Using
the property \(-|f(x)| \, ≤ \, f(x) \, ≤ \,|f(x)| \), we encompassed \(f(x)\) to ensure the function’s entirety is captured within bounds.
This property applied to functions translates to:

\[-\int_a^b |f(x)| \, dx \,≤\, \, ∫_a^b \,f(x) \, dx \, ≤ ∫_a^b |f(x)| \, dx\]
which then simplifies our understanding of spaces and bounds the values.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) bridges the concept of differentiation and integration. FTC has two parts:

• The first part states that if \(f\) is continuous over \([a, b]\) and \(F\) is an antiderivative of \(f\) on \([a, b]\), then:
  \[\textstyle \frac{d}{dx} \bigg (\bigintss_a^x f(t)\bigg) \bigg)=f(x)\]
• The second part states that if \(f\) is continuous over \([a, b]\), then:
  \( \bigintss_a^b f(x)dx=F(b)-F(a)\), where \(F\) is an antiderivative of \(f\).

In the context of our exercise, continuity of \(f\) ensures we can find antiderivatives and use integration properties directly. The FTC allows simplification of complex integral computations by recalling their fundamental principles.
Understanding this theorem helps recognize how integral values and their properties arise from derivatives, making it simpler to work with inequalities like \( |\bigintss_a^b f(x)dx |\bigintsleq \bigintss_a^b |f(x)|dx\).
The FTC provides the theoretical framework that integrates differentiation with integration, validating the steps in our exercise.