What is the volume of the region $S ={(x, y, z) : |x| + |y| + |z| ≤ 1}$?












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What is the volume of the region $S ={(x, y, z) : |x| + |y| + |z| ≤ 1}$ ?



How can i find the volume



any hints /solution






geometry multivariable-calculus volume







share|cite|improve this question











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    0












    $begingroup$


    What is the volume of the region $S ={(x, y, z) : |x| + |y| + |z| ≤ 1}$ ?



    How can i find the volume



    any hints /solution






    geometry multivariable-calculus volume







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      What is the volume of the region $S ={(x, y, z) : |x| + |y| + |z| ≤ 1}$ ?



      How can i find the volume



      any hints /solution






      geometry multivariable-calculus volume







      share|cite|improve this question











      $endgroup$




      What is the volume of the region $S ={(x, y, z) : |x| + |y| + |z| ≤ 1}$ ?



      How can i find the volume



      any hints /solution






      geometry multivariable-calculus volume




      geometry multivariable-calculus volume






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 6 at 7:56









      StubbornAtom

      6,28631339




      6,28631339










      asked Jan 6 at 3:25









      santoshsantosh

      989




      989






















          3 Answers
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          3












          $begingroup$

          The region $S$ looks somewhat like this:



          enter image description here



          If you consider only the first octant, you have the simplex:



          enter image description here



          Due to symmetry, the volume of $S$ is $8$ times the volume of this simplex.



          That is, $operatorname{vol}(S)=8times $$iiint_{substack{x+y+zle 1\ x,,y,,z,ge, 0}}mathrm{d}x,mathrm{d}y,mathrm{d}z$






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            First of all, you have to understand what $|x|+|y|+|z|=1$ means.



            Obviously this equation is fully symmetric when changing the octants. Thus one can restrict to the first one and we get then $x+y+z=1$ instead. But this is nothing but the plane through the points $X=(1,0,0)$, $Y=(0,1,0)$, and $Z=(0,0,1)$.



            Therefore your set $S$ is nothing but the octahedron, centered at the origin $O=(0,0,0)$, with circumradius $$r=overline{OX}=overline{OY}=overline{OZ}=1$$ i.e. with edge length $$s=overline{XY}=overline{XZ}=overline{YZ}=sqrt2$$



             

             

            Next one calculates the area $A_O$ of the triangle $OXY$ as $$A_O=frac12 overline{OX} overline{OY}=frac12$$ Then the volume $V$ of the octahedral octant $OXYZ$ would be $$V=frac13 A_O overline{OZ}=frac16$$ Thus the searched for volume of the full octahedron is $$vol(S)=8 V=frac86=frac43$$



             

             

            You even could derive the total surface area of that octahedron. Let $M$ be the center of $X$ and $Y$. Thus $M=(frac12, frac12, 0)$. Then you'll have the distances
            $$overline{XM}=overline{MY}=frac12 overline{XY}=frac{sqrt2}2=frac1{sqrt2}$$
            and
            $$overline{ZM}^2=overline{XZ}^2-overline{XM}^2=2-frac12=frac32$$
            Then the area $A_Z$ of the triangle $XYZ$ becomes
            $$A_Z=frac12 overline{XZ} overline{ZM}=overline{XM} overline{ZM}=frac1{sqrt2}cdot frac{sqrt3}{sqrt2}=frac{sqrt3}2$$
            and therefore finally
            $$textit{surf},(S)=8 A_Z=8cdotfrac{sqrt3}2=4sqrt3$$



            --- rk






            share|cite|improve this answer











            $endgroup$





















              0












              $begingroup$

              S is a diamond, to find the volume easily we can consider each octant, in which it is a tetrahedron of side length 1. So in each octant it has volume $frac{1}{6}$ and thus the overall volume is $frac43$.






              share|cite|improve this answer











              $endgroup$













              • $begingroup$
                @user376343 Good catch, just changed it!
                $endgroup$
                – Erik Parkinson
                Jan 6 at 5:31










              • $begingroup$
                The sides have to be specified, some of them have the length 1, but ..
                $endgroup$
                – user376343
                Jan 6 at 20:14











              Your Answer





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              3 Answers
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              3












              $begingroup$

              The region $S$ looks somewhat like this:



              enter image description here



              If you consider only the first octant, you have the simplex:



              enter image description here



              Due to symmetry, the volume of $S$ is $8$ times the volume of this simplex.



              That is, $operatorname{vol}(S)=8times $$iiint_{substack{x+y+zle 1\ x,,y,,z,ge, 0}}mathrm{d}x,mathrm{d}y,mathrm{d}z$






              share|cite|improve this answer









              $endgroup$


















                3












                $begingroup$

                The region $S$ looks somewhat like this:



                enter image description here



                If you consider only the first octant, you have the simplex:



                enter image description here



                Due to symmetry, the volume of $S$ is $8$ times the volume of this simplex.



                That is, $operatorname{vol}(S)=8times $$iiint_{substack{x+y+zle 1\ x,,y,,z,ge, 0}}mathrm{d}x,mathrm{d}y,mathrm{d}z$






                share|cite|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  The region $S$ looks somewhat like this:



                  enter image description here



                  If you consider only the first octant, you have the simplex:



                  enter image description here



                  Due to symmetry, the volume of $S$ is $8$ times the volume of this simplex.



                  That is, $operatorname{vol}(S)=8times $$iiint_{substack{x+y+zle 1\ x,,y,,z,ge, 0}}mathrm{d}x,mathrm{d}y,mathrm{d}z$






                  share|cite|improve this answer









                  $endgroup$



                  The region $S$ looks somewhat like this:



                  enter image description here



                  If you consider only the first octant, you have the simplex:



                  enter image description here



                  Due to symmetry, the volume of $S$ is $8$ times the volume of this simplex.



                  That is, $operatorname{vol}(S)=8times $$iiint_{substack{x+y+zle 1\ x,,y,,z,ge, 0}}mathrm{d}x,mathrm{d}y,mathrm{d}z$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 6 at 7:51









                  StubbornAtomStubbornAtom

                  6,28631339




                  6,28631339























                      3












                      $begingroup$

                      First of all, you have to understand what $|x|+|y|+|z|=1$ means.



                      Obviously this equation is fully symmetric when changing the octants. Thus one can restrict to the first one and we get then $x+y+z=1$ instead. But this is nothing but the plane through the points $X=(1,0,0)$, $Y=(0,1,0)$, and $Z=(0,0,1)$.



                      Therefore your set $S$ is nothing but the octahedron, centered at the origin $O=(0,0,0)$, with circumradius $$r=overline{OX}=overline{OY}=overline{OZ}=1$$ i.e. with edge length $$s=overline{XY}=overline{XZ}=overline{YZ}=sqrt2$$



                       

                       

                      Next one calculates the area $A_O$ of the triangle $OXY$ as $$A_O=frac12 overline{OX} overline{OY}=frac12$$ Then the volume $V$ of the octahedral octant $OXYZ$ would be $$V=frac13 A_O overline{OZ}=frac16$$ Thus the searched for volume of the full octahedron is $$vol(S)=8 V=frac86=frac43$$



                       

                       

                      You even could derive the total surface area of that octahedron. Let $M$ be the center of $X$ and $Y$. Thus $M=(frac12, frac12, 0)$. Then you'll have the distances
                      $$overline{XM}=overline{MY}=frac12 overline{XY}=frac{sqrt2}2=frac1{sqrt2}$$
                      and
                      $$overline{ZM}^2=overline{XZ}^2-overline{XM}^2=2-frac12=frac32$$
                      Then the area $A_Z$ of the triangle $XYZ$ becomes
                      $$A_Z=frac12 overline{XZ} overline{ZM}=overline{XM} overline{ZM}=frac1{sqrt2}cdot frac{sqrt3}{sqrt2}=frac{sqrt3}2$$
                      and therefore finally
                      $$textit{surf},(S)=8 A_Z=8cdotfrac{sqrt3}2=4sqrt3$$



                      --- rk






                      share|cite|improve this answer











                      $endgroup$


















                        3












                        $begingroup$

                        First of all, you have to understand what $|x|+|y|+|z|=1$ means.



                        Obviously this equation is fully symmetric when changing the octants. Thus one can restrict to the first one and we get then $x+y+z=1$ instead. But this is nothing but the plane through the points $X=(1,0,0)$, $Y=(0,1,0)$, and $Z=(0,0,1)$.



                        Therefore your set $S$ is nothing but the octahedron, centered at the origin $O=(0,0,0)$, with circumradius $$r=overline{OX}=overline{OY}=overline{OZ}=1$$ i.e. with edge length $$s=overline{XY}=overline{XZ}=overline{YZ}=sqrt2$$



                         

                         

                        Next one calculates the area $A_O$ of the triangle $OXY$ as $$A_O=frac12 overline{OX} overline{OY}=frac12$$ Then the volume $V$ of the octahedral octant $OXYZ$ would be $$V=frac13 A_O overline{OZ}=frac16$$ Thus the searched for volume of the full octahedron is $$vol(S)=8 V=frac86=frac43$$



                         

                         

                        You even could derive the total surface area of that octahedron. Let $M$ be the center of $X$ and $Y$. Thus $M=(frac12, frac12, 0)$. Then you'll have the distances
                        $$overline{XM}=overline{MY}=frac12 overline{XY}=frac{sqrt2}2=frac1{sqrt2}$$
                        and
                        $$overline{ZM}^2=overline{XZ}^2-overline{XM}^2=2-frac12=frac32$$
                        Then the area $A_Z$ of the triangle $XYZ$ becomes
                        $$A_Z=frac12 overline{XZ} overline{ZM}=overline{XM} overline{ZM}=frac1{sqrt2}cdot frac{sqrt3}{sqrt2}=frac{sqrt3}2$$
                        and therefore finally
                        $$textit{surf},(S)=8 A_Z=8cdotfrac{sqrt3}2=4sqrt3$$



                        --- rk






                        share|cite|improve this answer











                        $endgroup$
















                          3












                          3








                          3





                          $begingroup$

                          First of all, you have to understand what $|x|+|y|+|z|=1$ means.



                          Obviously this equation is fully symmetric when changing the octants. Thus one can restrict to the first one and we get then $x+y+z=1$ instead. But this is nothing but the plane through the points $X=(1,0,0)$, $Y=(0,1,0)$, and $Z=(0,0,1)$.



                          Therefore your set $S$ is nothing but the octahedron, centered at the origin $O=(0,0,0)$, with circumradius $$r=overline{OX}=overline{OY}=overline{OZ}=1$$ i.e. with edge length $$s=overline{XY}=overline{XZ}=overline{YZ}=sqrt2$$



                           

                           

                          Next one calculates the area $A_O$ of the triangle $OXY$ as $$A_O=frac12 overline{OX} overline{OY}=frac12$$ Then the volume $V$ of the octahedral octant $OXYZ$ would be $$V=frac13 A_O overline{OZ}=frac16$$ Thus the searched for volume of the full octahedron is $$vol(S)=8 V=frac86=frac43$$



                           

                           

                          You even could derive the total surface area of that octahedron. Let $M$ be the center of $X$ and $Y$. Thus $M=(frac12, frac12, 0)$. Then you'll have the distances
                          $$overline{XM}=overline{MY}=frac12 overline{XY}=frac{sqrt2}2=frac1{sqrt2}$$
                          and
                          $$overline{ZM}^2=overline{XZ}^2-overline{XM}^2=2-frac12=frac32$$
                          Then the area $A_Z$ of the triangle $XYZ$ becomes
                          $$A_Z=frac12 overline{XZ} overline{ZM}=overline{XM} overline{ZM}=frac1{sqrt2}cdot frac{sqrt3}{sqrt2}=frac{sqrt3}2$$
                          and therefore finally
                          $$textit{surf},(S)=8 A_Z=8cdotfrac{sqrt3}2=4sqrt3$$



                          --- rk






                          share|cite|improve this answer











                          $endgroup$



                          First of all, you have to understand what $|x|+|y|+|z|=1$ means.



                          Obviously this equation is fully symmetric when changing the octants. Thus one can restrict to the first one and we get then $x+y+z=1$ instead. But this is nothing but the plane through the points $X=(1,0,0)$, $Y=(0,1,0)$, and $Z=(0,0,1)$.



                          Therefore your set $S$ is nothing but the octahedron, centered at the origin $O=(0,0,0)$, with circumradius $$r=overline{OX}=overline{OY}=overline{OZ}=1$$ i.e. with edge length $$s=overline{XY}=overline{XZ}=overline{YZ}=sqrt2$$



                           

                           

                          Next one calculates the area $A_O$ of the triangle $OXY$ as $$A_O=frac12 overline{OX} overline{OY}=frac12$$ Then the volume $V$ of the octahedral octant $OXYZ$ would be $$V=frac13 A_O overline{OZ}=frac16$$ Thus the searched for volume of the full octahedron is $$vol(S)=8 V=frac86=frac43$$



                           

                           

                          You even could derive the total surface area of that octahedron. Let $M$ be the center of $X$ and $Y$. Thus $M=(frac12, frac12, 0)$. Then you'll have the distances
                          $$overline{XM}=overline{MY}=frac12 overline{XY}=frac{sqrt2}2=frac1{sqrt2}$$
                          and
                          $$overline{ZM}^2=overline{XZ}^2-overline{XM}^2=2-frac12=frac32$$
                          Then the area $A_Z$ of the triangle $XYZ$ becomes
                          $$A_Z=frac12 overline{XZ} overline{ZM}=overline{XM} overline{ZM}=frac1{sqrt2}cdot frac{sqrt3}{sqrt2}=frac{sqrt3}2$$
                          and therefore finally
                          $$textit{surf},(S)=8 A_Z=8cdotfrac{sqrt3}2=4sqrt3$$



                          --- rk







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Jan 7 at 21:13

























                          answered Jan 6 at 10:21









                          Dr. Richard KlitzingDr. Richard Klitzing

                          1,78016




                          1,78016























                              0












                              $begingroup$

                              S is a diamond, to find the volume easily we can consider each octant, in which it is a tetrahedron of side length 1. So in each octant it has volume $frac{1}{6}$ and thus the overall volume is $frac43$.






                              share|cite|improve this answer











                              $endgroup$













                              • $begingroup$
                                @user376343 Good catch, just changed it!
                                $endgroup$
                                – Erik Parkinson
                                Jan 6 at 5:31










                              • $begingroup$
                                The sides have to be specified, some of them have the length 1, but ..
                                $endgroup$
                                – user376343
                                Jan 6 at 20:14
















                              0












                              $begingroup$

                              S is a diamond, to find the volume easily we can consider each octant, in which it is a tetrahedron of side length 1. So in each octant it has volume $frac{1}{6}$ and thus the overall volume is $frac43$.






                              share|cite|improve this answer











                              $endgroup$













                              • $begingroup$
                                @user376343 Good catch, just changed it!
                                $endgroup$
                                – Erik Parkinson
                                Jan 6 at 5:31










                              • $begingroup$
                                The sides have to be specified, some of them have the length 1, but ..
                                $endgroup$
                                – user376343
                                Jan 6 at 20:14














                              0












                              0








                              0





                              $begingroup$

                              S is a diamond, to find the volume easily we can consider each octant, in which it is a tetrahedron of side length 1. So in each octant it has volume $frac{1}{6}$ and thus the overall volume is $frac43$.






                              share|cite|improve this answer











                              $endgroup$



                              S is a diamond, to find the volume easily we can consider each octant, in which it is a tetrahedron of side length 1. So in each octant it has volume $frac{1}{6}$ and thus the overall volume is $frac43$.







                              share|cite|improve this answer














                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited Jan 6 at 5:30

























                              answered Jan 6 at 3:31









                              Erik ParkinsonErik Parkinson

                              1,17519




                              1,17519












                              • $begingroup$
                                @user376343 Good catch, just changed it!
                                $endgroup$
                                – Erik Parkinson
                                Jan 6 at 5:31










                              • $begingroup$
                                The sides have to be specified, some of them have the length 1, but ..
                                $endgroup$
                                – user376343
                                Jan 6 at 20:14


















                              • $begingroup$
                                @user376343 Good catch, just changed it!
                                $endgroup$
                                – Erik Parkinson
                                Jan 6 at 5:31










                              • $begingroup$
                                The sides have to be specified, some of them have the length 1, but ..
                                $endgroup$
                                – user376343
                                Jan 6 at 20:14
















                              $begingroup$
                              @user376343 Good catch, just changed it!
                              $endgroup$
                              – Erik Parkinson
                              Jan 6 at 5:31




                              $begingroup$
                              @user376343 Good catch, just changed it!
                              $endgroup$
                              – Erik Parkinson
                              Jan 6 at 5:31












                              $begingroup$
                              The sides have to be specified, some of them have the length 1, but ..
                              $endgroup$
                              – user376343
                              Jan 6 at 20:14




                              $begingroup$
                              The sides have to be specified, some of them have the length 1, but ..
                              $endgroup$
                              – user376343
                              Jan 6 at 20:14


















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