Problem 6 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Zeigen Sie die Integrierbarkeit von \(f(x)=c\), \(c \in \mathbb{R}\), über \([a, b] .\) Was ergibt der Vergleich mit der elementaren Definition auf Seite \(94 ?\)

Step-by-Step Solution

Answer

The constant function \(f(x) = c\) is integrable over \([a, b]\), and the integral is \(c(b-a)\).

Step 1: Understanding the Function

The function given is a constant function, represented as \( f(x) = c \), where \( c \) is a real number. Here, we need to demonstrate that this function is integrable over the interval \([a, b]\). A constant function is continuous everywhere, thus integrable over any closed interval.

Step 2: Applying the Definition of Riemann Integrability

According to the Riemann integrability criterion, a bounded function on a closed interval is integrable if the set of discontinuities has measure zero. Since \( f(x) = c \) is continuous over \([a, b]\), there are no discontinuities, and hence it is integrable across the interval \([a, b]\).

Step 3: Computing the Integral

The integral of a constant function \(f(x) = c\) over an interval \([a, b]\) can be found using the formula:\[ \int_a^b c \, dx = c(b-a) \]This calculation follows from multiplying the constant \( c \) by the length of the interval \((b-a)\).

Step 4: Comparison with Elementary Definition

Page 94 likely discusses the fundamental theorem of calculus or basic integration concepts. According to these, the integral of a constant function \( c \) over \([a, b]\) is indeed \( c(b-a) \), matching our calculation. This reaffirms that the theory and the practice yield the same result for this type of function.

Key Concepts

Constant Function

A constant function is one where the output value remains the same regardless of the input. It is defined by the function equation \( f(x) = c \), where \( c \) is a constant real number. This means that no matter what value you plug in for \( x \), the function will always give you \( c \) as the output. Constant functions are straightforward because they create horizontal lines across the graph. This monotony is significant because it implies certain properties, such as continuity and integrability, which are crucial in calculus. Constant functions serve as a simple yet fundamental example for exploring more complex types of functions.

Continuous Function

In calculus, a continuous function is one that does not have any breaks, jumps, or holes. Essentially, you can draw its graph without lifting the pen from the paper. For a function like \( f(x) = c \), which is a constant function, it is inherently continuous over any interval because there are no points where the function is undefined or jumps to a different value.
Continuous functions are vital because they ensure the applicability of numerous mathematical theorems, including Riemann integration and the fundamental theorem of calculus. They provide smooth behavior, allowing for the computation of properties like derivatives and integrals, which are key in understanding the function's behavior over certain intervals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, two core aspects of calculus. It establishes that differentiation and integration are inverse operations. For a function constant over an interval, integrating it yields the area under the function's constant value, multiplied by the width of the interval. In our case, for \( f(x) = c \), the theorem supports the calculation:

• The integral of \( c \) over \([a, b]\) computes to \( c(b-a) \)
• This result concurs with the intuitive understanding that constant functions form rectangles with the base \( (b-a) \) and height \( c \)

The Fundamental Theorem justifies using such straightforward calculations while maintaining a robust mathematical foundation.

Integral Calculation

Integral calculation involves finding the area under the curve of a function across a specified interval. For a constant function like \( f(x) = c \), this is particularly simple. You can use the formula:\[\int_a^b c \, dx = c(b-a) \]This formula arises because the integral essentially multiplies the constant height \( c \) by the width of the interval \( (b-a) \). Since \( f(x) = c \) forms a flat line across the interval, calculating its integral involves multiplying this flat line's height by its width to find the enclosed area. Whether you're just beginning with integration or looking towards more complex applications, this integral serves as an excellent starting point for understanding the basic principles of integration.