The area bounded by the graph `y=|x-3|`, the x-axis and the lines `x=-2` and `x=3` is ([.] denotes the greatest integer function)

Options

A. 7 sq. units
B. 15 sq. units
C. 21 sq. units
D. 28 sq. units

Answer

To find the area bounded by the graph \( y = |x - 3| \), the x-axis, and the lines \( x = -2 \) and \( x = 3 \), we can follow these steps:

### Step 1: Understand the function
The function \( y = |x - 3| \) represents a V-shaped graph that opens upwards with its vertex at the point \( (3, 0) \). The function can be expressed as:
- \( y = 3 - x \) for \( x < 3 \)
- \( y = x - 3 \) for \( x \geq 3 \)

### Step 2: Determine the points of intersection with the x-axis
To find where the graph intersects the x-axis, we set \( y = 0 \):
\[ |x - 3| = 0 \implies x - 3 = 0 \implies x = 3 \]
Thus, the graph intersects the x-axis at \( (3, 0) \).

### Step 3: Identify the area of interest
We need to find the area between \( x = -2 \) and \( x = 3 \). Since \( x = -2 \) is less than \( 3 \), we will only consider the part of the graph from \( x = -2 \) to \( x = 3 \).

### Step 4: Evaluate the function on the interval
For \( x \) in the interval \( [-2, 3] \):
- When \( x < 3 \) (which is the case for our interval), we use \( y = 3 - x \).

### Step 5: Calculate the area under the curve
The area under the curve from \( x = -2 \) to \( x = 3 \) can be calculated using the definite integral:
\[ \text{Area} = \int_{-2}^{3} (3 - x) \, dx \]

### Step 6: Compute the integral
Calculating the integral:
\[ \int (3 - x) \, dx = 3x - \frac{x^2}{2} + C \]
Now, we evaluate it from \( -2 \) to \( 3 \):
\[ \text{Area} = \left[ 3(3) - \frac{(3)^2}{2} \right] - \left[ 3(-2) - \frac{(-2)^2}{2} \right] \]
Calculating the first part:
\[ 3(3) - \frac{9}{2} = 9 - 4.5 = 4.5 \]
Calculating the second part:
\[ 3(-2) - \frac{4}{2} = -6 - 2 = -8 \]
Now substituting back:
\[ \text{Area} = 4.5 - (-8) = 4.5 + 8 = 12.5 \]

### Step 7: Final area calculation
The area under the curve from \( x = -2 \) to \( x = 3 \) is \( 12.5 \) square units.

### Step 8: Apply the greatest integer function
Since the question asks for the area bounded by the graph, the final answer will be the greatest integer less than or equal to \( 12.5 \):
\[ \lfloor 12.5 \rfloor = 12 \]

### Final Answer
The area bounded by the graph \( y = |x - 3| \), the x-axis, and the lines \( x = -2 \) and \( x = 3 \) is \( 12 \).