How to solve the following ??

Question

How to solve the following ??

Consider a square with vertices at $(1, 1); (- 1, 1); (- 1, - 1); (1, - 1)$ $(1,1);(−1,1);(−1,−1);(1,−1)$ . Let $S$ $S$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Find the area of region $S$ $S$ .

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Answer 1 · 2017-10-06T14:14:49

See below.

Explanation:

The sought region is the interior of the boundaries given by

$\sqrt{x^{2} + y^{2}} \leq x + 1$ $sqrt(x^2+y^2) le x+1$
$\sqrt{x^{2} + y^{2}} \leq 1 - x$ $sqrt(x^2+y^2) le 1-x$
$\sqrt{x^{2} + y^{2}} \leq y + 1$ $sqrt(x^2+y^2) le y+1$
$\sqrt{x^{2} + y^{2}} \leq 1 - y$ $sqrt(x^2+y^2) le 1-y$

The area computation is left as an exercise.

Attached a region plot

enter image source here

NOTE:

The intersection region boundary point at the first quadrant is

$\sqrt{2} - 1, \sqrt{2} - 1$ $sqrt2-1,sqrt2-1$ and the area is given by

$S = {(2 (\sqrt{2} - 1))}^{2} + 4 (\int_{1 - \sqrt{2}}^{\sqrt{2} - 1} \frac{1 - x^{2}}{2} d x - 2 {(\sqrt{2} - 1)}^{2}) = \frac{4}{3} (4 \sqrt{2} - 5)$ $S= (2 (sqrt[2] - 1))^2 +4(int_(1-sqrt2)^(sqrt2-1)(1-x^2)/2 dx - 2(sqrt2-1)^2) = 4/3(4sqrt2-5)$

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How to solve the following ??

Consider a square with vertices at $(1, 1); (- 1, 1); (- 1, - 1); (1, - 1)$ $(1,1);(−1,1);(−1,−1);(1,−1)$ . Let $S$ $S$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Find the area of region $S$ $S$ .

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

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How to solve the following ??

Consider a square with vertices at (1,1);(−1,1);(−1,−1);(1,−1)(1,1);(−1,1);(−1,−1);(1,−1). Let SS be the region consisting of all points inside the square which are nearer to the origin than to any edge. Find the area of region SS.

1 Answer

Explanation:

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Impact of this question

Consider a square with vertices at $(1, 1); (- 1, 1); (- 1, - 1); (1, - 1)$ $(1,1);(−1,1);(−1,−1);(1,−1)$ . Let $S$ $S$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Find the area of region $S$ $S$ .