Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Find \(\int_{-1}^{1}|x| d x\) using the area interpretation of the definite integral. (b) Show that \(\frac{\left|x^{2}\right|}{2}\) is not an antiderivative of \(|x|\) on \([-1,1]\) by showing that applying the Fundamental Theorem as if it were, gives the wrong answer. (c) Find an antiderivative of \(|x|\) on \([-1,0)\). Find an antiderivative of \(|x|\) on \((0,1]\).

Step-by-Step Solution

Answer

\(\int_{-1}^{1}|x| d x = 1\) ; \(\frac{|x^2|}{2}\) is not an antiderivative of \(|x|\) on \([-1,1]\) ; Antiderivatives of \(|x|\) are \(-\frac{x^2}{2}\) on \([-1,0)\) and \(\frac{x^2}{2}\) on \((0,1]\)

Step 1: Evaluate \(\int_{-1}^{1}|x| d x\) using area interpretation

The absolute value of \(x\) from \(-1\) to \(1\) forms two triangles with base \(1\) and height \(1\). The area under the curve of each triangle is thus \(\frac{1}{2} * base * height = \frac{1}{2}\). As there are two triangles, add them together to get the integral. Therefore, \(\int_{-1}^{1}|x| d x = 2 * \frac{1}{2} = 1\).

Step 2: Show that \(\frac{|x^2|}{2}\) is not an antiderivative of \(|x|\)

To show this, apply the fundamental theorem as if \(\frac{|x^2|}{2}\) were the antiderivative of \(|x|\). The definite integral from \(-1\) to \(1\) of \(|x|\) should be equal to \(\frac{|x^2|}{2}\) evaluated at \(1\) minus \(\frac{|x^2|}{2}\) evaluated at \(-1\). Which would result in \( \frac{1}{2} - \left(-\frac{1}{2}\right) = 1\). However, this is the wrong answer as from step 1 we know the integral should be \(1\). Therefore, \(\frac{|x^2|}{2}\) is not the antiderivative of \(|x|\).

Step 3: Find the antiderivatives of \(|x|\) on \([-1,0)\) and \((0,1]\)

For \(x < 0\), \(|x| = -x\), so its antiderivative is \(-\frac{x^2}{2}\). For \(x > 0\), \(|x| = x\), so its antiderivative is \(\frac{x^2}{2}\).

Key Concepts

Area Interpretation of Definite Integrals

Definite integrals have a fascinating connection to geometry, particularly when we consider the area under a curve. In basic terms, the definite integral of a function \( f(x) \) from \( a \) to \( b \), written as \( \int_{a}^{b} f(x) \, dx \), can be understood as the area between the graph of \( f(x) \) and the x-axis over the interval \( [a, b] \). This forms the basis for the "area interpretation" of integrals. When dealing with the integral \( \int_{-1}^{1} |x| \, dx \) as in our exercise, we employ this interpretation. The graph of \( |x| \) forms two triangles: one from \( x = -1 \) to \( x = 0 \) and the other from \( x = 0 \) to \( x = 1 \). Both triangles have a base and height of \( 1 \).

• Triangle on \( x < 0 \): Base = 1, Height = 1, Area = \( \frac{1}{2} \)
• Triangle on \( x > 0 \): Base = 1, Height = 1, Area = \( \frac{1}{2} \)

Adding these areas gives a total integral value of 1. This illustrates how definite integrals can be solved through geometrical insights.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of differentiation and integration, providing a way to evaluate definite integrals. It states that if \( F(x) \) is an antiderivative of \( f(x) \) on an interval \( [a, b] \), then the definite integral of \( f(x) \) from \( a \) to \( b \) is given by \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a). \]In the exercise, \( \frac{|x^2|}{2} \) is examined to determine if it's an antiderivative of \( |x| \). We apply the theorem:

• Compute \( \frac{|x^2|}{2} \) at \( x = 1 \): \( \frac{1}{2} \)
• Compute \( \frac{|x^2|}{2} \) at \( x = -1 \): \(-\frac{1}{2} \)
• Subtract: \( \frac{1}{2} - (-\frac{1}{2}) = 1 \)

While it matches the area calculated, using this specific antiderivative for \( |x| \) is just a coincidence for this particular interval as it doesn't work for others. Hence, \( \frac{|x^2|}{2} \) isn't generally valid across the interval, showcasing the necessity of finding a correct antiderivative.

Antiderivatives of the Absolute Value Function

Antiderivatives are functions which "reverse" differentiation. Finding an antiderivative of \( |x| \) requires examining the behavior of \( |x| \) on different intervals. Since \(|x|\) behaves differently on either side of zero, we split into two parts:For \( x < 0 \): Here, \( |x| = -x \). So, the antiderivative is:

• If \( f(x) = -x \), then the antiderivative is \( F(x) = -\frac{x^2}{2} + C \), where \( C \) is a constant of integration.

On \( x > 0 \): For this part, \( |x| = x \). The antiderivative becomes:

• If \( f(x) = x \), then the antiderivative is \( F(x) = \frac{x^2}{2} + C \).

Each provides the "area under the curve" in a specific region, showing how antiderivatives are crucial in integrating non-standard functions like absolute values.