The area bounded by the curves `y=|x|-1a n dy=-|x|+1` is 1 sq. units (b) 2 sq. units `2sqrt(2)` sq. units (d) 4 sq. units

Options

A. `alpha=e^(2)+1`
B. `alpha=e^(2)-2`
C. `beta=1+e^(-1)`
D. `beta=1+e^(-2)`

Explanation

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

### Step 1: Identify the equations of the curves
The curves given are:
1. \( y = |x| - 1 \)
2. \( y = -|x| + 1 \)

### Step 2: Determine the points of intersection
To find the points where these two curves intersect, we set them equal to each other:

\[ |x| - 1 = -|x| + 1 \]

This can be split into two cases based on the definition of the absolute value.

**Case 1:** \( x \geq 0 \)

In this case, \( |x| = x \), so the equation becomes:

\[ x - 1 = -x + 1 \]
\[ 2x = 2 \implies x = 1 \]

**Case 2:** \( x < 0 \)

Here, \( |x| = -x \), so the equation becomes:

\[ -x - 1 = x + 1 \]
\[ -2x = 2 \implies x = -1 \]

Thus, the points of intersection are \( (1, 0) \) and \( (-1, 0) \).

### Step 3: Sketch the curves
- For \( y = |x| - 1 \):
- When \( x \geq 0 \), it is a line with a slope of 1 starting from the point (0, -1).
- When \( x < 0 \), it is a line with a slope of -1 starting from the point (0, -1).

- For \( y = -|x| + 1 \):
- When \( x \geq 0 \), it is a line with a slope of -1 starting from the point (0, 1).
- When \( x < 0 \), it is a line with a slope of 1 starting from the point (0, 1).

### Step 4: Find the area between the curves
The area can be calculated by integrating the difference between the two functions from \( x = -1 \) to \( x = 1 \):

\[ \text{Area} = \int_{-1}^{1} \left((-|x| + 1) - (|x| - 1)\right) \, dx \]

This simplifies to:

\[ \text{Area} = \int_{-1}^{1} \left(-|x| + 1 - |x| + 1\right) \, dx = \int_{-1}^{1} (2 - 2|x|) \, dx \]

### Step 5: Calculate the integral
Since the function is symmetric about the y-axis, we can calculate the area from 0 to 1 and then double it:

\[ \text{Area} = 2 \int_{0}^{1} (2 - 2x) \, dx \]

Calculating the integral:

\[ = 2 \left[ 2x - x^2 \right]_{0}^{1} = 2 \left[ (2 \cdot 1 - 1^2) - (2 \cdot 0 - 0^2) \right] = 2(2 - 1) = 2 \]

### Final Answer
Thus, the area bounded by the curves is \( 2 \) square units.

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