Key Concepts and Formulas

• Area Under a Curve: The area bounded by a curve $y = f(x)$, the x-axis, and the lines $x = a$ and $x = b$ is given by $\int_a^b |f(x)| dx$.
• Absolute Value Function: $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.
• Definite Integral Properties: $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ for $a < c < b$.

Step-by-Step Solution

Step 1: Define the absolute value function piecewise. We need to express $|x - 2|$ as a piecewise function to handle the absolute value.

$$|x - 2| = \begin{cases} x - 2, & \text{if } x \ge 2 \\ -(x - 2) = 2 - x, & \text{if } x < 2 \end{cases}$$

Reasoning: This step is necessary to remove the absolute value and express the function in a form suitable for integration.

Step 2: Set up the integral for the area. The area is given by the integral of $|x - 2|$ from $x = 1$ to $x = 3$. Since the function changes its form at $x = 2$, we split the integral into two parts.

$$\text{Area} = \int_1^3 |x - 2| dx = \int_1^2 (2 - x) dx + \int_2^3 (x - 2) dx$$

Reasoning: This divides the area calculation into regions where the absolute value function has a consistent form.

Step 3: Evaluate the first integral.

$$\int_1^2 (2 - x) dx = \left[2x - \frac{x^2}{2}\right]_1^2 = \left(2(2) - \frac{2^2}{2}\right) - \left(2(1) - \frac{1^2}{2}\right) = (4 - 2) - \left(2 - \frac{1}{2}\right) = 2 - \frac{3}{2} = \frac{1}{2}$$

Reasoning: We apply the power rule for integration and then evaluate at the limits of integration.

Step 4: Evaluate the second integral.

$$\int_2^3 (x - 2) dx = \left[\frac{x^2}{2} - 2x\right]_2^3 = \left(\frac{3^2}{2} - 2(3)\right) - \left(\frac{2^2}{2} - 2(2)\right) = \left(\frac{9}{2} - 6\right) - (2 - 4) = \left(\frac{9}{2} - \frac{12}{2}\right) - (-2) = -\frac{3}{2} + 2 = \frac{1}{2}$$

Reasoning: We apply the power rule for integration and then evaluate at the limits of integration.

Step 5: Calculate the total area.

$$\text{Area} = \frac{1}{2} + \frac{1}{2} = 1$$

Reasoning: We add the areas from the two integrals to get the total area.

Common Mistakes & Tips

• Forgetting the absolute value: Failing to account for the absolute value can lead to incorrect signs and an incorrect area calculation.
• Splitting the integral: Remember to split the integral at the point where the expression inside the absolute value changes sign.
• Careful with signs: Pay close attention to signs when evaluating the definite integrals, especially after substituting the limits.

Summary

We calculated the area bounded by $y = |x - 2|$, $x = 1$, $x = 3$, and the x-axis by splitting the integral into two parts, one from $x = 1$ to $x = 2$ and the other from $x = 2$ to $x = 3$. We then evaluated each integral separately and added the results to obtain the total area, which is 1. This corresponds to option (D).

Final Answer

The final answer is \boxed{1}, which corresponds to option (D).