Question

Let \[ S = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \leq 1\}. \] Then the area of \( S \) equals ...............

Hint:

For geometric shapes like squares or diamonds, the area can be easily calculated by determining the side lengths and using basic area formulas.

Answer 1

Correct Answer: 1.9 - 2.1

The set
\[S={(x,y)\in\mathbb{R}^2:\ |x|+|y|\le 1}\]
represents a diamond-shaped region.

Step 1: Geometry of (S)

The boundary \(|x|+|y|=1\) is a square (diamond) with vertices
\[(1,0),\ (0,1),\ (-1,0),\ (0,-1).\]

The diagonals of this square have lengths:
\[d_1=2 \quad \text{(horizontal)}, \qquad d_2=2 \quad \text{(vertical)}.\]

Step 2: Area

The area of a rhombus with diagonals \(d_1\) and \(d_2\) is
\[\text{Area}=\frac{1}{2}d_1d_2 =\frac{1}{2}\cdot2\cdot2 =2.\]

\[\boxed{\text{Area}(S)=2}\]