Find area enclosed by `|x|+|y|=1`.

Answer

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

### Step 1: Understand the Equation
The equation \( |x| + |y| = 1 \) describes a geometric figure in the coordinate plane. It represents a diamond (or rhombus) shape centered at the origin.

### Step 2: Determine the Quadrants
We will analyze the equation in each quadrant:
- **First Quadrant**: \( x \geq 0 \) and \( y \geq 0 \) leads to \( x + y = 1 \).
- **Second Quadrant**: \( x \leq 0 \) and \( y \geq 0 \) leads to \( -x + y = 1 \) or \( y = 1 + x \).
- **Third Quadrant**: \( x \leq 0 \) and \( y \leq 0 \) leads to \( -x - y = 1 \) or \( y = -1 - x \).
- **Fourth Quadrant**: \( x \geq 0 \) and \( y \leq 0 \) leads to \( x - y = 1 \) or \( y = x - 1 \).

### Step 3: Plot the Lines
Now we can plot these lines:
- The line \( x + y = 1 \) intersects the axes at (1,0) and (0,1).
- The line \( -x + y = 1 \) intersects the axes at (-1,0) and (0,1).
- The line \( -x - y = 1 \) intersects the axes at (-1,0) and (0,-1).
- The line \( x - y = 1 \) intersects the axes at (1,0) and (0,-1).

### Step 4: Identify the Vertices of the Diamond
The vertices of the diamond are:
- \( (1, 0) \)
- \( (0, 1) \)
- \( (-1, 0) \)
- \( (0, -1) \)

### Step 5: Calculate the Area
The diamond can be divided into four triangles, each having a base and height of 1 unit. The area of one triangle can be calculated as:
\[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \]
Since there are four such triangles, the total area enclosed by the diamond is:
\[ \text{Total Area} = 4 \times \frac{1}{2} = 2 \text{ square units} \]

### Final Answer
Thus, the area enclosed by the curve \( |x| + |y| = 1 \) is \( 2 \) square units.

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