Finding volume of the region enclosed by x 2 + y 2 − 6 x = 0 , z = 36 − x 2 − y 2 z = 0I'm trying to calculate the volume of the region enclosed by the cylinder x 2 + y 2 − 6 x = 0, the semicircle z = 36 − x 2 − y 2 and the plane z = 0. I tried moving the region towards -x so that the cylinder has it's center at zero. I ended up using these equations and proposing this integral using cylindrical coordinates:New cylinder x 2 + y 2 = 9New semicircle z = 36 − ( x + 3 ) 2 − y 2 ∫ 0 2 π ∫ 0 3 ∫ 0 36 − ( r c o s ( θ ) + 3 ) 2 − ( r s i n ( θ ) ) 2 r ⋅ d z d r d θFor some reason I can't solve it this way and I don't know what am I doing wrong.

Question

Finding volume of the region enclosed by       x    2    +      y    2    −  6  x  =  0  ,  z  =      36    −          x      2        −          y      2        z  =  0I&#039;m trying to calculate the volume of the region enclosed by the cylinder       x    2    +      y    2    −  6  x  =  0, the semicircle   z  =      36    −          x      2        −          y      2       and the plane   z  =  0. I tried moving the region towards -x so that the cylinder has it&#039;s center at zero. I ended up using these equations and proposing this integral using cylindrical coordinates:New cylinder       x    2    +      y    2    =  9New semicircle   z  =      36    −    (    x    +    3          )      2        −          y      2            ∫          0              2      π            ∫          0              3            ∫          0                      36        −        (        r        c        o        s        (        θ        )        +        3                  )          2                −        (        r        s        i        n        (        θ        )                  )          2                      r  ⋅  d  z  d  r  d  θFor some reason I can&#039;t solve it this way and I don&#039;t know what am I doing wrong.

sweetwisdomgw · Accepted Answer

Explanation:If you take cylindrical coordinates   x  =  r  cos  ⁡  θ  ,  y  =  r  sin  ⁡  θ  ,  z  =  z with Jacobian   J  =  r, then you&#039;ll have       ∫          0              2      π            ∫          0              6      cos      ⁡      θ            ∫          0                      36        −                  r          2                      r  d  θ  d  r  d  z

Find volume of the region enclosed by x^2+y^2-6x=0, z and =sqrt{36-x^2-y^2}, z=0.

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