Find the area of the region bounded by the curve `y=|x+1|`, lines `x= -4,x=2` and X-axis.

Options

A. `5`
B. `7`
C. `6`
D. `9`

Explanation

To find the area of the region bounded by the curve \( y = |x + 1| \), the lines \( x = -4 \), \( x = 2 \), and the x-axis, we will follow these steps:

### Step 1: Understand the function \( y = |x + 1| \)

The function \( y = |x + 1| \) can be expressed in piecewise form:
- 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

We need to find the points where the curve intersects the x-axis:
- Set \( y = 0 \):
\[ |x + 1| = 0 \implies x + 1 = 0 \implies x = -1 \]

### Step 3: Determine the area segments

We will calculate the area in two segments:
1. From \( x = -4 \) to \( x = -1 \) (where \( y = -x - 1 \))
2. From \( x = -1 \) to \( x = 2 \) (where \( y = x + 1 \))

### Step 4: Set up the integrals for the area

The area \( A \) can be calculated as:
\[ A = \int_{-4}^{-1} (-x - 1) \, dx + \int_{-1}^{2} (x + 1) \, dx \]

### Step 5: Calculate the first integral

Calculate \( \int_{-4}^{-1} (-x - 1) \, dx \):
\[ \int (-x - 1) \, dx = -\frac{x^2}{2} - x \]
Now evaluate from \( -4 \) to \( -1 \):
\[ = \left[-\frac{(-1)^2}{2} - (-1)\right] - \left[-\frac{(-4)^2}{2} - (-4)\right] \]
\[ = \left[-\frac{1}{2} + 1\right] - \left[-\frac{16}{2} + 4\right] \]
\[ = \left[\frac{1}{2}\right] - \left[-8 + 4\right] \]
\[ = \frac{1}{2} - (-4) = \frac{1}{2} + 4 = \frac{1}{2} + \frac{8}{2} = \frac{9}{2} \]

### Step 6: Calculate the second integral

Calculate \( \int_{-1}^{2} (x + 1) \, dx \):
\[ \int (x + 1) \, dx = \frac{x^2}{2} + x \]
Now evaluate from \( -1 \) to \( 2 \):
\[ = \left[\frac{(2)^2}{2} + 2\right] - \left[\frac{(-1)^2}{2} + (-1)\right] \]
\[ = \left[2 + 2\right] - \left[\frac{1}{2} - 1\right] \]
\[ = 4 - \left[\frac{1}{2} - \frac{2}{2}\right] = 4 - \left[-\frac{1}{2}\right] = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \]

### Step 7: Combine the areas

Now, add the two areas together:
\[ A = \frac{9}{2} + \frac{9}{2} = \frac{18}{2} = 9 \]

### Final Answer

The area of the region bounded by the curve \( y = |x + 1| \), the lines \( x = -4 \), \( x = 2 \), and the x-axis is \( \boxed{9} \).