Question

Find area enclosed by $[|x|]+[|y|] = 2024$, [.]=GIF.

Answer 1

8100

Answer 2

The given equation is $[|x|]+[|y|] = k$, where $k = 2024$. Here, $[.]$ denotes the Greatest Integer Function (GIF).

Let $I_x = [|x|]$ and $I_y = [|y|]$. Since $I_x$ and $I_y$ are results of the greatest integer function applied to non-negative values ($|x|$ and $|y|$), $I_x$ and $I_y$ must be non-negative integers. So, we are looking for pairs of non-negative integers $(I_x, I_y)$ such that $I_x + I_y = k$.

The possible pairs $(I_x, I_y)$ are: $(0, k), (1, k-1), (2, k-2), \dots, (k-1, 1), (k, 0)$. The number of such pairs is $k+1$.

For each pair $(I_x, I_y)$, the equation defines a region in the xy-plane: $I_x \le |x| < I_x+1$ AND $I_y \le |y| < I_y+1$.

1. For $x$:
   • If $I_x = 0$: $0 \le |x| < 1 \implies -1 < x < 1$. The length of this interval is $1 - (-1) = 2$.
   • If $I_x > 0$: $I_x \le |x| < I_x+1 \implies (I_x \le x < I_x+1)$ or $(-(I_x+1) < x \le -I_x)$. The length of the interval $[I_x, I_x+1)$ is 1. The length of the interval $(-(I_x+1), -I_x]$ is 1. The total length of the x-region is $1+1=2$.

In both cases ($I_x=0$ or $I_x>0$), the total length of the x-region is 2.

2. For $y$:
   • Similarly, if $I_y = 0$: $0 \le |y| < 1 \implies -1 < y < 1$. The length of this interval is $1 - (-1) = 2$.
   • If $I_y > 0$: $I_y \le |y| < I_y+1 \implies (I_y \le y < I_y+1)$ or $(-(I_y+1) < y \le -I_y)$. The length of the interval $[I_y, I_y+1)$ is 1. The length of the interval $(-(I_y+1), -I_y]$ is 1. The total length of the y-region is $1+1=2$.

In both cases ($I_y=0$ or $I_y>0$), the total length of the y-region is 2.

For each pair $(I_x, I_y)$, the region defined is a rectangle (or a union of rectangles) with dimensions 2 by 2. The area of each such region $R_{I_x, I_y}$ is $2 \times 2 = 4$.

Since there are $k+1$ such disjoint regions, each with an area of 4, the total enclosed area is the sum of the areas of these regions. Total Area = (Number of pairs) $\times$ (Area of each region) = $(k+1) \times 4 = 4k+4$.

Substitute $k=2024$: Total Area = $4(2024) + 4 = 8096 + 4 = 8100$.

The area enclosed by $[|x|]+[|y|] = 2024$ is 8100 square units.