Step 1: Understanding the Concept:
The area bounded by a function \( y = f(x) \) and the x-axis between \( x=a \) and \( x=b \) is given by \( \int_{a}^{b} |f(x)| dx \).
Step 2: Key Formula or Approach:
Identify if the function crosses the x-axis in the interval \([-1, 2]\).
\( -\frac{2}{3}x + 2 = 0 \implies x = 3 \).
Since \( x=3 \) is outside \([-1, 2]\), the function does not cross the axis in this interval.
Step 3: Detailed Explanation:
\[ Area = \int_{-1}^{2} \left( -\frac{2}{3}x + 2 \right) dx \]
\[ = \left[ -\frac{2}{3} \cdot \frac{x^2}{2} + 2x \right]_{-1}^{2} = \left[ -\frac{x^2}{3} + 2x \right]_{-1}^{2} \]
\[ = \left( -\frac{4}{3} + 4 \right) - \left( -\frac{1}{3} - 2 \right) \]
\[ = \left( \frac{8}{3} \right) - \left( -\frac{7}{3} \right) = \frac{15}{3} = 5 \]
(Self-Correction: Re-calculating: \( 8/3 + 7/3 = 15/3 = 5 \). The integral calculation: \( (-4/3 + 4) = 8/3 \). \( (-1/3 - 2) = -7/3 \). The difference is \( 15/3 = 5 \). Given the options, let's re-read: maybe \( 13/3 \)? No, integral is correct.)
Step 4: Final Answer:
The calculated area is 5.