Integrating over or under this objectThere is clearly a kink in my understanding of double integrals with polar coordinates.Our problem is to find the volume enclosed by the hyperboloid − x 2 − y 2 + z 2 = 1 and plane z = 2. I correctly figure the bounds to be D = { ( r , θ ) | 0 ≤ θ ≤ 2 π , 0 ≤ r ≤ 3 } .But I have trouble finding if we are integrating between the section of the hyperboloid and the disk r = 1 at z = 0 or z = 2.While r 1 − r 2 d r d θ would make some sense as an integrand, I essentially dont know if I am integrating (ie, finding volume) under or over the surface of the hyperboloid.

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Integrating over or under this objectThere is clearly a kink in my understanding of double integrals with polar coordinates.Our problem is to find the volume enclosed by the hyperboloid   −      x    2    −      y    2    +      z    2    =  1 and plane   z  =  2. I correctly figure the bounds to be   D  =  {  (  r  ,  θ  )      |    0  ≤  θ  ≤  2  π  ,  0  ≤  r  ≤      3    }  .But I have trouble finding if we are integrating between the section of the hyperboloid and the disk   r  =  1 at   z  =  0 or   z  =  2.While   r      1    −          r      2        d  r  d  θ would make some sense as an integrand, I essentially dont know if I am integrating (ie, finding volume) under or over the surface of the hyperboloid.

blogpolisft · Accepted Answer

Step 1Since you&#039;ve said in the comments that you wish to find the volume, this can be done using a triple integral. The region in the xy plane can be written in polar coordinates as the set D you have said. As for the z limits, we&#039;ll have       1    +          x      2        +          y      2        ≤  z  ≤  2, therefore  V  =  ∭  d  V  =      ∫    0                  3                  ∫    0          2      π            ∫                  1        +                  r          2                      2    r    d  z    d  θ    d  r  .Step 2This ensures that you are computing the volume between the hyperboloid and the plane   z  =  2.In general when you compute volumes you compute from the xy plane up to whatever height you wish. When we set       1    +          x      2        +          y      2        ≤  z  ≤  2 we are essentially doing two computations: the volume of the cylinder of height 2 and subtracting the volume up to the hyperboloid, giving the volume desired.

Find the volume enclosed by the hyperboloid -x^2-y^2+z^2=1 and plane z=2. I correctly figure the bounds to be D={(r,theta)|0 leq theta leq 2pi, 0 leq r leq sqrt(3)}

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