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The volume of the region $S=\{(x,y,z) : \mid x \mid +\mid y \mid + \mid z \mid \leq 1\}$ is

1. $\frac{1}{6}\\$
2. $\frac{1}{3}\\$
3. $\frac{2}{3}\\$
4. $\frac{4}{3}$

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Option D

We know that $|x| = x$ when $x>= 0$ and $|x| = -x$ when $x<0$.

So, region covered by the graph $|x| + |y| <= 1$ is bounded by the lines $(x+y =1)$ , $(-x +y =1)$ , $(x - y =1)$ , $(-x -y =1)$ [As shown in Fig $A$ below]

So, the region covered will be the square $[ (0,1) , (1,0) , (0,-1) , (-1,0) ]$ [ As shown in Fig $A$]

The side length of the above square is $√2.$

Now, in the $3D$ version of the above scenario the space will be covered by the lines $(x+z = 1)$ , $( -x +z = 1)$ , $( -y +z =1)$ ,

$( y +z = 1)$, $( -y-z =1)$, $( -y +z = 1)$ , $(x -z =1)$ , $(-x -z = 1)$. [ As shown in the figure $B$]

This will create two pyramids $[ (0,0,1) , (0,1,0) , (1,0,0) , (0,-1,0) , (-1,0,0) ]$ and $[ (0,0,-1) , (0,1,0) , (1,0,0) , (0,-1,0) , (-1,0,0) ]$

[ As shown in the above figure $B$ ]

Note that the base of the above two pyramids are same and it is the square formed in the $2D$ version.

Now, we know that the volume of the pyramid is $1/3$ ( area of the base) (height)

Here the area of the base $\left ( \sqrt{2} \right )^{2}$  $=$ $2$ square units. and the height = $1$ unit.

Thus the volume of the pyramids formed $=$ $2$ * ( $1/3$ * $2$ * $1$ ) cubic units = $4/3$ cubic units

which matches ith option $D$.

Hence, $D$ will be the correct answer.

by Loyal (9.8k points)
edited