The area bounded by the line y = x, X-axis and the lines x = -1, x = 2 is

Options

A. `5/2 sq. units`
B. `3/2 sq. units`
C. `1/3 sq. units`
D. `2/3 sq. units`

Answer

To find the area bounded by the line \( y = x \), the x-axis, and the vertical lines \( x = -1 \) and \( x = 2 \), we can follow these steps:

### Step 1: Understand the Boundaries
The area we need to calculate is bounded by:
- The line \( y = x \)
- The x-axis (which is \( y = 0 \))
- The vertical lines \( x = -1 \) and \( x = 2 \)

### Step 2: Sketch the Graph
Draw the line \( y = x \). It will intersect the x-axis at the origin (0,0). The vertical lines \( x = -1 \) and \( x = 2 \) will help define the area we are interested in.

### Step 3: Set Up the Integral
The area can be divided into two parts:
1. From \( x = -1 \) to \( x = 0 \)
2. From \( x = 0 \) to \( x = 2 \)

For the first part (from \( x = -1 \) to \( x = 0 \)), the upper function is the x-axis (y=0) and the lower function is the line \( y = x \). Thus, the area can be expressed as:
\[ \text{Area}_1 = \int_{-1}^{0} (0 - x) \, dx = \int_{-1}^{0} -x \, dx \]

For the second part (from \( x = 0 \) to \( x = 2 \)), the upper function is the line \( y = x \) and the lower function is the x-axis (y=0). Thus, the area can be expressed as:
\[ \text{Area}_2 = \int_{0}^{2} (x - 0) \, dx = \int_{0}^{2} x \, dx \]

### Step 4: Calculate the Integrals
Now we will calculate both integrals.

**For Area 1:**
\[ \text{Area}_1 = \int_{-1}^{0} -x \, dx = \left[-\frac{x^2}{2}\right]_{-1}^{0} = \left[-\frac{0^2}{2}\right] - \left[-\frac{(-1)^2}{2}\right] = 0 - \left[-\frac{1}{2}\right] = \frac{1}{2} \]

**For Area 2:**
\[ \text{Area}_2 = \int_{0}^{2} x \, dx = \left[\frac{x^2}{2}\right]_{0}^{2} = \left[\frac{2^2}{2}\right] - \left[\frac{0^2}{2}\right] = \frac{4}{2} - 0 = 2 \]

### Step 5: Add the Areas
Now, we can find the total area by adding both areas:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]

### Final Answer
The area bounded by the line \( y = x \), the x-axis, and the lines \( x = -1 \) and \( x = 2 \) is:
\[ \frac{5}{2} \]