Area enclosed by equation `y = |x| -1 and y =1 - |x|` is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To find the area enclosed by the equations \( y = |x| - 1 \) and \( y = 1 - |x| \), we can follow these steps:

### Step 1: Identify the equations
We have two equations:
1. \( y = |x| - 1 \)
2. \( y = 1 - |x| \)

### Step 2: Find the points of intersection
To find the area enclosed by these two curves, we first need to determine where they intersect. We set the equations equal to each other:

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

### Step 3: Solve for \( x \)
We will solve this equation by considering two cases for \( |x| \).

**Case 1:** \( x \geq 0 \)
In this case, \( |x| = x \), so the equation becomes:
\[ x - 1 = 1 - x \]
Adding \( x \) to both sides gives:
\[ 2x - 1 = 1 \]
Adding 1 to both sides results in:
\[ 2x = 2 \implies x = 1 \]

**Case 2:** \( x < 0 \)
Here, \( |x| = -x \), so the equation becomes:
\[ -x - 1 = 1 + x \]
Adding \( x \) to both sides gives:
\[ -x - 1 - x = 1 \implies -2x - 1 = 1 \]
Adding 1 to both sides results in:
\[ -2x = 2 \implies x = -1 \]

### Step 4: Find corresponding \( y \) values
Now, we find the \( y \) values at the points of intersection:
- For \( x = 1 \):
\[ y = |1| - 1 = 1 - 1 = 0 \]
- For \( x = -1 \):
\[ y = |-1| - 1 = 1 - 1 = 0 \]

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

### Step 5: Determine the area enclosed
The area can be calculated by integrating the difference of the two functions from \( -1 \) to \( 1 \):

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

This simplifies to:
\[ \text{Area} = \int_{-1}^{1} (2 - 2|x|) \, dx \]

### Step 6: Evaluate the integral
We can split the integral into two parts, from \( -1 \) to \( 0 \) and from \( 0 \) to \( 1 \):

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

Calculating the first integral:
\[ \int_{-1}^{0} (2 + 2x) \, dx = \left[ 2x + x^2 \right]_{-1}^{0} = (0 + 0) - (-2 + 1) = 1 \]

Calculating the second integral:
\[ \int_{0}^{1} (2 - 2x) \, dx = \left[ 2x - x^2 \right]_{0}^{1} = (2 - 1) - (0 - 0) = 1 \]

Thus, the total area is:
\[ \text{Area} = 1 + 1 = 2 \text{ square units} \]

### Final Answer
The area enclosed by the equations \( y = |x| - 1 \) and \( y = 1 - |x| \) is \( 2 \) square units.

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