Computing volume of a set in R 3 Let Ω = { ( x , y , z ) : x 2 + y 2 ≤ z 2 ≤ 4 }.So, the problem is to find the volume of this set. MY though: comput ∭ Ω d VBut, im having trouble seeing the picture of the set, and hence finding limit of integration. Can someone help me?

Question

Computing volume of a set in             R        3  Let   Ω  =  {  (  x  ,  y  ,  z  )  :      x    2    +      y    2    ≤      z    2    ≤  4  }.So, the problem is to find the volume of this set. MY though: comput       ∭          Ω        d  VBut, im having trouble seeing the picture of the set, and hence finding limit of integration. Can someone help me?

afinat4s · Accepted Answer

Step 1Another way to compute this is using integration. Since   Ω  =  {  (  x  ,  y  ,  z  )        |          x    2    +      y    2    ≤      z    2    ≤  4  }.Consider the upper cone (in the sense of   z  ≥  0. Then, on the upper cone,             x      2        +          y      2        ≤  z  ≤  2,   −      4    −          x      2        ≤  y  ≤      4    −          x      2      ,   −  2  ≤  x  ≤  2      ∭          Ω          d  V  =  2      ∫          −      2        2        ∫          −              4        −                  x          2                                    4        −                  x          2                          ∫                            x          2                +                  y          2                      2    1     d  z    d  y    d  xStep 2which upon changing to cylindrical coordinates is indeed      ∭    Ω    d  V  =  2      ∫    0          2      π            ∫    0    2        ∫    r    2    r    d  z    d  r    d  θ  =            16      π        3

Janiya Rose · Accepted Answer

Explanation:It&#039;s a double cone with an axis Z of height 2 and with a radius of the base 2. Hence   V  =  2  π      r    2        h    3    =  2  π  ∗  4  ∗      2    3    =      16    3    π

Find the volume of this set. Let Omega={(x,y,z):x^2+y^2 leq z^2 leq 4}

Open question

Answer & Explanation

New Questions in High school geometry