Finding the volume below the part of the plane which is above the xy-planeMy function is f ( x , y ) = 2 − | x | − | y | and I'm supposed to find the volume below the part of the plane which is above the xy-plane.I don't understand how to find the limits of my integrals for this problem. I have tried to draw the lines for all the cases for which the absolute value of x and y is both positive and negative, but I don't see the limits. Is there anyone who has any suggestions?

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Finding the volume below the part of the plane which is above the xy-planeMy function is   f  (  x  ,  y  )  =  2  −      |    x      |    −      |    y      |   and I&#039;m supposed to find the volume below the part of the plane which is above the xy-plane.I don&#039;t understand how to find the limits of my integrals for this problem. I have tried to draw the lines for all the cases for which the absolute value of x and y is both positive and negative, but I don&#039;t see the limits. Is there anyone who has any suggestions?

Lillianna Salazar · Accepted Answer

Explanation:The graph of this function is not a plane. However, this graph is symmetric with respect to the plane   x  =  0 (  f  (  x  ,  y  )  =  f  (  x  ,  −  y  )) and the plane   y  =  0 (  f  (  x  ,  y  )  =  f  (  −  x  ,  y  )). Hence it suffices to consider the case where    x  ≥  0 and   y  ≥  0 and multiply the result by 4:  V  =  4      ∫          {      x      ≥      0      ,      y      ≥      0      ,      x      +      y      ≤      2      }        (  2  −  x  −  y  )    d  x  d  y  =  4      ∫          x      =      0        2        (          ∫              y        =        0                    2        −        x              (    2    −    x    −    y    )        d    y    )      d  x  .

Kameron Wang · Accepted Answer

Step 1For example, let us take the planes      {                            z          =          2          −          x          −          y                                      z          =          2          −          x          +          y                            ⟹    y  =  0  =        the                 x      z            −         plane  Thus, projecting on the xy- plane, we get      {                            y          =          2          −          x                                      y          =          −          2          +          x                            ⟹    2  x  =  4    ⟹    x  =  2    ,    y  =  0Step 2Observe that for the whole four planes, you get as projection a square on the xy- plane with vertices (2, 0), (-2,0) and etc.

My function is f(x,y)=2-|x|-|y| and I'm supposed to find the volume below the part of the plane which is above the xy-plane.

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