Finding the Area Bounded by |x| + |y| = 1

The question asks for the area bounded by the curve defined by the equation $|x| + |y| = 1$. This equation represents a specific shape in the Cartesian coordinate system.

To understand the shape, let's analyze the equation in different quadrants based on the signs of x and y:

• Quadrant I (x ≥ 0, y ≥ 0): The equation becomes \(x + y = 1\). This is a straight line.
• Quadrant II (x < 0, y ≥ 0): The equation becomes \(-x + y = 1\). This is a straight line.
• Quadrant III (x < 0, y < 0): The equation becomes \(-x - y = 1\), which can be written as \(x + y = -1\). This is a straight line.
• Quadrant IV (x ≥ 0, y < 0): The equation becomes \(x - y = 1\). This is a straight line.

Let's find the points where this shape intersects the axes:

• If \(y = 0\), then \(|x| + |0| = 1\), which means \(|x| = 1\). So, \(x = 1\) or \(x = -1\). The points are \((1, 0)\) and \((-1, 0)\).
• If \(x = 0\), then \(|0| + |y| = 1\), which means \(|y| = 1\). So, \(y = 1\) or \(y = -1\). The points are \((0, 1)\) and \((0, -1)\).

These four points are the vertices of the shape. Plotting these points and connecting them with lines according to the equations in each quadrant reveals that the shape bounded by \(|x| + |y| = 1\) is a square with vertices at \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\).

We can calculate the area of this square using several methods.

Method 1: Using side length

Let's find the length of one side of the square. Consider the side connecting \((1, 0)\) and \((0, 1)\). Using the distance formula, the length \(s\) is:

\(s = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}\)

The area of a square is \(s^2\).

\(\text{Area} = (\sqrt{2})^2 = 2 \text{ square units}\)

Method 2: Using diagonals

The shape is a square whose diagonals lie along the x and y axes. The vertices are \((1, 0)\), \((-1, 0)\), \((0, 1)\), \((0, -1)\). The length of the diagonal along the x-axis is the distance between \((1, 0)\) and \((-1, 0)\), which is \(1 - (-1) = 2\). The length of the diagonal along the y-axis is the distance between \((0, 1)\) and \((0, -1)\), which is \(1 - (-1) = 2\). The diagonals of a square are equal in length.

The area of a square can also be calculated using the formula \(\text{Area} = \frac{1}{2} \times d^2\), where \(d\) is the length of the diagonal.

\(\text{Area} = \frac{1}{2} \times (2)^2 = \frac{1}{2} \times 4 = 2 \text{ square units}\)

Method 3: Sum of areas of triangles

The square is made up of four right-angled triangles, one in each quadrant, with their vertices at the origin \((0, 0)\) and the axis intercepts. For example, in the first quadrant, the vertices are \((0, 0)\), \((1, 0)\), and \((0, 1)\). This is a right-angled triangle with base 1 and height 1.

The area of one such triangle is \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = 0.5 \text{ square units}\).

Since there are four such triangles forming the square, the total area is \(4 \times 0.5 = 2 \text{ square units}\).

All methods yield the same result.

The area bounded by the curve \(|x| + |y| = 1\) is 2 square units.

Revision Table: Key Concepts for Area Calculation

Additional Information: Related Absolute Value Graphs

Understanding the graph of \(|x| + |y| = k\) is useful. For any positive constant \(k\), the graph of \(|x| + |y| = k\) is a square centered at the origin with vertices at \((k, 0)\), \((-k, 0)\), \((0, k)\), and \((0, -k)\). The length of the diagonal is \(2k\).

Other related graphs include:

• \(|x| = k\): Represents two vertical lines, \(x = k\) and \(x = -k\).
• \(|y| = k\): Represents two horizontal lines, \(y = k\) and \(y = -k\).
• \(|x| = |y|\): Represents the lines \(y = x\) and \(y = -x\).

The equation \(|x| + |y| = 1\) is a specific case of \(|x| + |y| = k\) where \(k=1\).

The area bounded by \(|x| + |y| = k\) is \(\frac{1}{2} \times (2k)^2 = \frac{1}{2} \times 4k^2 = 2k^2\). For \(k=1\), the area is \(2(1)^2 = 2\).