Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate \(\int|x| d x\).

Step-by-Step Solution

Answer

The integral evaluates to infinity over an infinite domain.

Step 1: Understanding Absolute Value Function

The function \(|x|\) represents the absolute value of \(x\), which is \(-x\) for \(x < 0\) and \(x\) for \(x \geq 0\). This means we need to consider the behavior of the function in both quadrants (negative and non-negative).

Step 2: Splitting the Integral

To solve the integral of the absolute value function, we split it into two parts based on the domain: \[ \int |x| \, dx = \int_{-\infty}^0 (-x) \, dx + \int_0^{\infty} x \, dx. \] This effectively accounts for the change in behavior of the absolute value function.

Step 3: Solving the Integral for Negative Values

For \(x < 0\), the function \(-x\) is integrated as follows: \[ \int_{-\infty}^0 (-x) \, dx = \int_{-\infty}^0 -x \, dx = -\frac{x^2}{2} \bigg|_{-\infty}^0 = 0. \] This integral evaluates to zero because the \(x^2\) term dominates as \(x \to -\infty\), and at \(x = 0\) it also contributes zero.

Step 4: Solving the Integral for Non-Negative Values

For \(x \geq 0\), the function \(x\) is integrated as follows: \[ \int_0^{\infty} x \, dx = \frac{x^2}{2} \bigg|_0^{\infty}. \] This integral evaluates to \(\infty\), representing the area under the curve \(y = x\) from zero to infinity.

Step 5: Combining the Results

Combining both parts of the integral, we have: \[ \int_{-\infty}^0 (-x) \, dx + \int_0^{\infty} x \, dx = 0 + \infty = \infty. \] Therefore, the integral \(\int |x| \, dx\) over its entire domain is divergent and evaluates to \(\infty\), suggesting it cannot be calculated in a standard sense over infinite limits.

Key Concepts

Absolute Value Function

The absolute value function \(|x|\) is one of the simplest yet most important concepts in calculus. It expresses the non-negative magnitude of a real number, meaning that it tells you how far away a number is from zero without considering direction. In terms of mathematics, the absolute value is defined as follows:

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

This piecewise function presents a change in the definition depending on the sign of the input. Hence, when dealing with the integral of \(|x|\), it's crucial to account for this switch in behavior, splitting the integral into parts where the function has different expressions.

Improper Integrals

Improper integrals occur when we integrate over an unbounded interval, such as from \(-\infty\) to \(\infty\), or when the function itself is unbounded in the range of integration. In the given exercise, the integration of \(|x|\) is improper because it examines the entire real number line. Evaluating an improper integral often involves limits, and it can result in divergent outcomes—those that do not produce a finite number. This exercise demonstrates such divergence, as the total area tends to infinity due to the nature of the linear function \(x\) over infinite limits.

Integration Techniques

The integration of functions, particularly those that come from piecewise definitions like an absolute value, can require special techniques. For simple linear parts defined by \(-x\) and \(x\), traditional techniques such as basic antiderivatives apply:

• The antiderivative of \(-x\) is \(-\frac{x^2}{2}\).
• The antiderivative of \(x\) is \(\frac{x^2}{2}\).

These techniques help reduce the complex integral into simpler pieces, yet due to the infinite nature of the bounds, they result in divergent integrals unless further restricted to finite intervals. Mastering when and how to apply these strategies is essential for correctly evaluating integrals involving absolute value functions.

Splitting Integrals

Splitting integrals is a strategy used when dealing with piecewise functions, such as \(|x|\). Since this function has two different forms over different intervals, we need to split the integral accordingly. This means creating separate integrals:

• One for \(x < 0\), where the function behaves as \(-x\).
• Another for \(x \geq 0\), where it behaves as \(x\).

By solving these integrals individually, we can account for the different behaviors of the absolute value function. This technique ensures each piece is treated correctly according to its definition before attempting to combine the results, often unveiling behaviors such as divergence in improper integrals.