Find the volume bounded by the following surfacesI need to find the volume bounded by the following surfaces: x 2 + y 2 = 2 z x 2 + y 2 = 3 − zI don't know how to proceed in solving this exercise, all I could think of was trying to make a system of these 2 and get z = 1. I just wanna understand how am I supposed to proceed in finding the boundaries, no need to evaluate the integral.

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Find the volume bounded by the following surfacesI need to find the volume bounded by the following surfaces:      x    2    +      y    2    =  2  z      x    2    +      y    2    =  3  −  zI don&#039;t know how to proceed in solving this exercise, all I could think of was trying to make a system of these 2 and get   z  =  1. I just wanna understand how am I supposed to proceed in finding the boundaries, no need to evaluate the integral.

cydostwng6c · Accepted Answer

Step 1You have       x    2    +      y    2    =  2  (  3  −      x    2    −      y    2    )    ⟹        x    2    +      y    2    =  6  −  2      x    2    −  2      y    2      ⟹    3      x    2    +  3      y    2    =  6      x    2    +      y    2    =  2, which gives you the circle of center (0,0) and radius       2  . We can now compute the integral in cylindrical coordinates doing the following calculations      ∭    V    d  x  d  y  d  z  =      ∫    0          2      π            ∫    0                  2                  ∫                            ρ          2                2                    3      −              ρ        2              ρ  d  z  d  ρ  d  θ  =      ∫    0          2      π        d  θ      ∫    0                  2              ρ            (        3  −            3              ρ        2              2              )        d  ρ  =  =  2  π      ∫    0                  2              3  ρ  −            3              ρ        3              2    d  ρ  =  2  π  ⋅            3              ρ        2              2    −            3              ρ        4              8                      |              0                  2              =  3  π  .

I need to find the volume bounded by the following surfaces: x^2+y^2=2z, x^2+y^2=3-z.

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