Volume of a Hyperboloid using triple integration/shadow methodCould someone help me find the volume of the hyperboloid x 2 1817 + y 2 1817 − z 2 10914 = 1with the limits in the z − a x i s of -130 to 43? I tried using triple integration, but I'm not sure how to convert the variable the in limit of the innermost integral into a number. I would also be open to any other methods of finding the volume.

Question

Volume of a Hyperboloid using triple integration/shadow methodCould someone help me find the volume of the hyperboloid            x      2        1817    +            y      2        1817    −            z      2        10914    =  1with the limits in the   z  −  a  x  i  s of -130 to 43? I tried using triple integration, but I&#039;m not sure how to convert the variable the in limit of the innermost integral into a number. I would also be open to any other methods of finding the volume.

Adelyn Mercado · Accepted Answer

Step 1Usually triple integration is used to find weight of a volume with the density function d(x,y,z). However you can still adjust your density   d  (  x  ,  y  ,  z  )  =  1 in order to obtain the volume.I suggest you to draw the volume by a mathematical tool to see the natural bounds before integration.I think the double integral you are using is on XY-plane. Thus in order to define limits of integration you should put   z  =  0. Then you get a circle equation             x      2        1817    +            y      2        1817    =  1Step 2Then if your differential is dxdy then the x values should be inside  [  −      (    1817  −      y    2    )  ,      (    1817  −      y    2    )  ]The outer integration bound (which bounds y values) is  [  −      (    1817  )  ,      (    1817  )  ]Finally you will have;      ∫          −              (            1817      )                      (            1817      )            ∫          −              (            1817      −              y        2            )                      (            1817      −              y        2            )        F  (  x  ,  y  )  .  d  x  d  yNote that as you have limits on z-axis, you will arrange the integrand function   z  =  F  (  x  ,  y  ) by these limits when the z-value crosses them.

Find the volume of the hyperboloid x^2/1817+y^2/1817-z^2/10914=1 with the limits in the z-axis of -130 to 43?

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