Problem with finding the volume with cylindrical coordinates.I have the following function: Z = x 2 + y 2 , Z = x + ythat i want to solve (to find the volume) with cylindrical coordinates. I am evaluating the integral to get: V = ∭ d V = ∬ R ∫ x 2 + y 2 x + y d z d A = ∬ R x + y − ( x 2 + y 2 ) d A .and from here i am trying to get the bounds for r by intersecting the two functions and i get that r = 0 o r r = 1 therefore i tought that 0 ≤ r ≤ 1 but it dosen't seems right. But i get the following result: 2 π ∫ 0 1 ( r − r 2 ) r d r d θ = 2 π ( r 3 3 − r 4 4 ) | 0 1 = π 6 and this is not the corrent answer, i should get π 8 .. I know this is too specific to a given problem question i apologize for that but can someone tells me where i made a mistake

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Problem with finding the volume with cylindrical coordinates.I have the following function:  Z  =      x    2    +      y    2    ,     Z  =  x  +  ythat i want to solve (to find the volume) with cylindrical coordinates. I am evaluating the integral to get:  V  =  ∭  d  V  =      ∬          R              ∫                            x          2                +                  y          2                            x      +      y          d  z    d  A  =      ∬          R        x  +  y  −  (      x    2    +      y    2    )    d  A  .and from here i am trying to get the bounds for r by intersecting the two functions and i get that   r  =  0     o  r     r  =  1 therefore i tought that   0  ≤  r  ≤  1 but it dosen&#039;t seems right. But i get the following result:  2  π      ∫          0              1        (  r  −      r    2    )  r  d  r  d  θ  =  2  π            (                  r      3        3    −            r      4        4              )                          |              0    1    =      π    6  and this is not the corrent answer, i should get       π    8    .. I know this is too specific to a given problem question i apologize for that but can someone tells me where i made a mistake

decoratesuw · Accepted Answer

Step 1In cylindrical coordinates, you&#039;re after the volume of the region between   z  =      r    2   and   z  =  r            (        cos  ⁡  (  θ  )  +  sin  ⁡  (  θ  )            )      . Note that       r    2    ⩽  r            (        cos  ⁡  (  θ  )  +  sin  ⁡  (  θ  )            )          ⟺    r  ⩽  cos  ⁡  (  θ  )  +  sin  ⁡  (  θ  )  .Step 2So, you want to have   cos  ⁡  (  θ  )  +  sin  ⁡  (  θ  )  ⩾  0. That means that you have to compute       ∫          −      π              /            4              3      π              /            4            ∫    0          cos      ⁡      (      θ      )      +      sin      ⁡      (      θ      )            ∫                  r        2                    r      (      cos      ⁡      (      θ      )      +      sin      ⁡      (      θ      )      )        r        d    z        d    r        d    θ  , which is indeed equal to             π      8

Following function: Z=x^2+y^2, Z=x+y that i want to solve (to find the volume) with cylindrical coordinates.

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