Problem 79 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \(\frac{1}{2} x|x|\) is an antiderivative of \(|x|\), and use this fact to get a simple formula for \(\int_{a}^{b}|x| d x\).

Step-by-Step Solution

Answer

\(\int_{a}^{b}|x| dx = \frac{1}{2} b|b| - \frac{1}{2} a|a|\) using the antiderivative \(\frac{1}{2} x|x|\).

Step 1: Differentiate the Antiderivative

To prove that \(\frac{1}{2} x|x|\) is an antiderivative of \(|x|\), differentiate \(\frac{1}{2} x|x|\) with respect to \(x\). Note that \(|x|\) is a piecewise function: \(|x| = x \) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).

Step 2: Handle the Case for \(x \geq 0\)

For \(x \geq 0\), \(|x| = x\), so \(\frac{1}{2} x|x| = \frac{1}{2} x^2\). The derivative \(\frac{d}{dx}\left( \frac{1}{2} x^2 \right) = x\).

Step 3: Handle the Case for \(x < 0\)

For \(x < 0\), \(|x| = -x\), so \(\frac{1}{2} x|x| = \frac{1}{2} x(-x) = -\frac{1}{2} x^2\). The derivative \(\frac{d}{dx}\left(-\frac{1}{2} x^2 \right) = -x\), which equals \(|x|\) when \(x < 0\).

Step 4: Conclusion of Differentiation

Since for both cases (\(x \geq 0\) and \(x < 0\)), the derivative of \(\frac{1}{2} x|x|\) is \(|x|\), \(\frac{1}{2} x|x|\) is indeed an antiderivative of \(|x|\).

Step 5: Set Up the Definite Integral

To find the integral \(\int_{a}^{b}|x| dx\), use the antiderivative \(F(x) = \frac{1}{2} x|x|\). Apply the Fundamental Theorem of Calculus: \(\int_{a}^{b} |x| \, dx = F(b) - F(a)\).

Step 6: Evaluate the Integral

Substitute in the antiderivative and evaluate: \(F(b) = \frac{1}{2} b |b|\) and \(F(a) = \frac{1}{2} a |a|\). Thus, \(\int_{a}^{b} |x| \, dx = \frac{1}{2} b |b| - \frac{1}{2} a |a|\).

Key Concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), is a fundamental mathematical concept that represents the distance of a number from zero on the number line. It is always non-negative, whether the input value is positive or negative. Understanding how \(|x|\) works is crucial, as it is often used in calculus and algebra.

The absolute function is defined piecewise as follows:

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

Essentially, the absolute value of a number equals the number itself if it's positive or zero, and the negation of the number if it is negative.

In calculus, working with absolute values can involve considering separate cases, especially when deriving functions or calculating integrals like the exercise at hand.

Piecewise Functions

Piecewise functions are mathematical expressions that have different definitions over different parts of their domain. This is significant when dealing with functions like the absolute value \(|x|\) that behave differently depending on the input value.

Understanding piecewise functions involves recognizing that the function is not continuous over its entire domain and must be evaluated in these distinct segments:

• For \(x \geq 0\), \(|x|\) behaves as \(x\).
• For \(x < 0\), \(|x|\) is \(-x\).

When performing calculus operations on a piecewise function, like finding an antiderivative, each segment must be handled separately due to its distinct characteristics. Such piecewise handling ensures correct computation of derivatives or integrals, especially around points where the function's behavior changes.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, two core operations in calculus. It has two main parts that provide a bridge between the antiderivative and the definite integral.

For the problem at hand, the theorem is used to evaluate a definite integral by utilizing the antiderivative. The theorem states that if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\).

This principle enables the computation of the area under the curve of \(f(x)\) from \(a\) to \(b\) using the values of the antiderivative at these points, which simplifies evaluations involving complex functions such as piecewise definitions.

Definite Integral

The definite integral is a central concept in calculus that allows for the calculation of the total accumulation of quantities, such as area under a curve, over a specific interval. Unlike an indefinite integral, which produces a general antiderivative, the definite integral computes a specific numerical value.

In the problem given, we evaluate the integral of the absolute value function \(\int_{a}^{b}|x| \, dx\) using its antiderivative \(\frac{1}{2}x|x|\). The Fundamental Theorem of Calculus plays a crucial role here by providing us with the formula:

• Evaluate \(F(b) = \frac{1}{2} b |b|\)
• Subtract the value \(F(a) = \frac{1}{2} a |a|\)
• The result gives \(\int_{a}^{b} |x| \, dx = \frac{1}{2} b |b| - \frac{1}{2} a |a|\)

Through these calculations, the definite integral provides a straightforward way to quantify the area under the curve represented by \(|x|\) over the specified intervals.