Finding the volume of f ( x , y , z ) = z inside the cylinder and outside the hyperboloidI need to setup the triple integral in cartesian coordinates to solve the volume of f ( x , y , z ) = z inside the cylinder x 2 + y 2 = 10 and outside the hyperboloid x 2 + y 2 − z 2 = 1 in the first octant.What I did so far is to find the intersection between the cylinder and hyperboloid so that I can find the bounds for z. Actually, I am uncertain if finding this intersection is a right procedure. 10 − z 2 = 1 ⟹ z = 3My bounds for z is 0 ≤ z ≤ 3 since the volume is also bounded by the first octant. My bounds for y is 1 − x 2 ≤ y ≤ 10 − x 2 . Lastly, my bounds for x is 1 ≤ x ≤ 10 . My iterated integral for this one will be like this ∫ 1 10 ∫ 1 − x 2 10 − x 2 ∫ 0 3 z d z d y d xDo I miss something here?

Question

Finding the volume of   f  (  x  ,  y  ,  z  )  =  z inside the cylinder and outside the hyperboloidI need to setup the triple integral in cartesian coordinates to solve the volume of   f  (  x  ,  y  ,  z  )  =  z inside the cylinder       x    2    +      y    2    =  10 and outside the hyperboloid       x    2    +      y    2    −      z    2    =  1 in the first octant.What I did so far is to find the intersection between the cylinder and hyperboloid so that I can find the bounds for z. Actually, I am uncertain if finding this intersection is a right procedure.  10  −      z    2    =  1    ⟹    z  =  3My bounds for z is   0  ≤  z  ≤  3 since the volume is also bounded by the first octant. My bounds for y is       1    −          x      2        ≤  y  ≤      10    −          x      2      . Lastly, my bounds for x is   1  ≤  x  ≤      10  . My iterated integral for this one will be like this       ∫          1                      10                  ∫                  1        −                  x          2                                    10        −                  x          2                          ∫          0              3        z  d  z  d  y  d  xDo I miss something here?

jbacapzh · Accepted Answer

Step 1No that is not correct. Both z bounds cannot be constant.It is much easier in cylindrical coordinates but if you have to set it up in cartesian coordinates,Integrating wrt z first,   0  ≤  z  ≤            x      2        +          y      2        −    1  .1) For   0  ≤  x  ≤  1, y is bound between two circles in XY-plane (      x    2    +      y    2    =  1 and       x    2    +      y    2    =  10).      1    −          x      2        ≤  y  ≤      10    −          x      2      Step 22) For   1  ≤  x  ≤      10  , y  is bound between x-axis and circle       x    2    +      y    2    =  10.So the integral would be,             ∫      0      1              ∫                        1          −                      x            2                                                10          −                      x            2                                      ∫      0                                    x            2                    +                      y            2                    −          1                      z         d    z         d    y         d    x              +              ∫      1                        10                            ∫      0                        10          −                      x            2                                      ∫      0                                    x            2                    +                      y            2                    −          1                      z         d    z         d    y         d    x

I need to setup the triple integral in cartesian coordinates to solve the volume of f(x, y, z)=z inside the cylinder x^2+y^2=10 and outside the hyperboloid x^2+y^2-z^2=1 in the first octant.

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