Using intergration, find the area of the region bounded by the lines `y=|x+1|,x= -3, x=1` and X-axis.

Answer

To find the area of the region bounded by the lines \( y = |x + 1| \), \( x = -3 \), \( x = 1 \), and the x-axis, we will follow these steps:

### Step 1: Understand the function \( y = |x + 1| \)
The function \( y = |x + 1| \) can be split into two cases based on the value of \( x \):
- For \( x + 1 \geq 0 \) (i.e., \( x \geq -1 \)), \( y = x + 1 \).
- For \( x + 1 < 0 \) (i.e., \( x < -1 \)), \( y = -(x + 1) = -x - 1 \).

### Step 2: Identify the points of intersection
The lines \( x = -3 \) and \( x = 1 \) will serve as the vertical boundaries for our area. We also need to find where the function intersects the x-axis (i.e., where \( y = 0 \)):
- Setting \( |x + 1| = 0 \) gives \( x + 1 = 0 \) or \( x = -1 \).

### Step 3: Determine the area segments
The area can be divided into two segments:
1. From \( x = -3 \) to \( x = -1 \): Here, \( y = -x - 1 \).
2. From \( x = -1 \) to \( x = 1 \): Here, \( y = x + 1 \).

### Step 4: Calculate the area for each segment using integration
**Area from \( x = -3 \) to \( x = -1 \):**
\[ \text{Area}_1 = \int_{-3}^{-1} (-x - 1) \, dx \]
Calculating this integral:
\[ = \int_{-3}^{-1} (-x - 1) \, dx = \left[-\frac{x^2}{2} - x\right]_{-3}^{-1} \]
Calculating the limits:
\[ = \left[-\frac{(-1)^2}{2} - (-1)\right] - \left[-\frac{(-3)^2}{2} - (-3)\right] \]
\[ = \left[-\frac{1}{2} + 1\right] - \left[-\frac{9}{2} + 3\right] \]
\[ = \left[\frac{1}{2}\right] - \left[-\frac{3}{2}\right] = \frac{1}{2} + \frac{3}{2} = 2 \]

**Area from \( x = -1 \) to \( x = 1 \):**
\[ \text{Area}_2 = \int_{-1}^{1} (x + 1) \, dx \]
Calculating this integral:
\[ = \left[\frac{x^2}{2} + x\right]_{-1}^{1} \]
Calculating the limits:
\[ = \left[\frac{(1)^2}{2} + (1)\right] - \left[\frac{(-1)^2}{2} + (-1)\right] \]
\[ = \left[\frac{1}{2} + 1\right] - \left[\frac{1}{2} - 1\right] \]
\[ = \left[\frac{3}{2}\right] - \left[-\frac{1}{2}\right] = \frac{3}{2} + \frac{1}{2} = 2 \]

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

### Final Answer
The area of the region bounded by the lines \( y = |x + 1| \), \( x = -3 \), \( x = 1 \), and the x-axis is \( \boxed{4} \) square units.