How to find the spherical coordinates limits calculate the volume of the solid region?












0












$begingroup$


The solid bounded below by the hemisphere $rho=1,$ $ 0leq z$ and above by the cardoid of revolution $rho=1+cosphi$
I am new to the triple integral in spherical coordinates.
I know that limits of $rho$ will be $1leqrholeq 1+cosphi$, the limits on $theta$ will be $0leqthetaleq2pi$. So the integral will be
$$int_{0}^{2pi}int_{?}^{?}int_{1}^{1+cosphi}(rho)^2sinphi drho dphi dtheta$$
But how to find the limits of $phi$ in this case, I don't know.
Can anyone help!
Thanks in advance!






multivariable-calculus multiple-integral







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    0












    $begingroup$


    The solid bounded below by the hemisphere $rho=1,$ $ 0leq z$ and above by the cardoid of revolution $rho=1+cosphi$
    I am new to the triple integral in spherical coordinates.
    I know that limits of $rho$ will be $1leqrholeq 1+cosphi$, the limits on $theta$ will be $0leqthetaleq2pi$. So the integral will be
    $$int_{0}^{2pi}int_{?}^{?}int_{1}^{1+cosphi}(rho)^2sinphi drho dphi dtheta$$
    But how to find the limits of $phi$ in this case, I don't know.
    Can anyone help!
    Thanks in advance!






    multivariable-calculus multiple-integral







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      The solid bounded below by the hemisphere $rho=1,$ $ 0leq z$ and above by the cardoid of revolution $rho=1+cosphi$
      I am new to the triple integral in spherical coordinates.
      I know that limits of $rho$ will be $1leqrholeq 1+cosphi$, the limits on $theta$ will be $0leqthetaleq2pi$. So the integral will be
      $$int_{0}^{2pi}int_{?}^{?}int_{1}^{1+cosphi}(rho)^2sinphi drho dphi dtheta$$
      But how to find the limits of $phi$ in this case, I don't know.
      Can anyone help!
      Thanks in advance!






      multivariable-calculus multiple-integral







      share|cite|improve this question











      $endgroup$




      The solid bounded below by the hemisphere $rho=1,$ $ 0leq z$ and above by the cardoid of revolution $rho=1+cosphi$
      I am new to the triple integral in spherical coordinates.
      I know that limits of $rho$ will be $1leqrholeq 1+cosphi$, the limits on $theta$ will be $0leqthetaleq2pi$. So the integral will be
      $$int_{0}^{2pi}int_{?}^{?}int_{1}^{1+cosphi}(rho)^2sinphi drho dphi dtheta$$
      But how to find the limits of $phi$ in this case, I don't know.
      Can anyone help!
      Thanks in advance!






      multivariable-calculus multiple-integral




      multivariable-calculus multiple-integral






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      share|cite|improve this question













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      edited Jan 7 at 17:29









      Shubham Johri

      4,885717




      4,885717










      asked Jan 7 at 17:05









      Noor AslamNoor Aslam

      15112




      15112






















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          $begingroup$

          Notice that for the cardioid of revolution, $rhoge1$ for $0lephilepi/2$ and $rho<1$ for $pi/2<philepi$. It intersects the hemisphere only in the $xy$ plane. Therefore, $0lephilepi/2$.






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          • $begingroup$
            Sir the limits on the $rho$ and $theta$ I have written are correct?
            $endgroup$
            – Noor Aslam
            Jan 8 at 12:47










          • $begingroup$
            @NoorAslam They are correct
            $endgroup$
            – Shubham Johri
            Jan 8 at 18:01











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          $begingroup$

          Notice that for the cardioid of revolution, $rhoge1$ for $0lephilepi/2$ and $rho<1$ for $pi/2<philepi$. It intersects the hemisphere only in the $xy$ plane. Therefore, $0lephilepi/2$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Sir the limits on the $rho$ and $theta$ I have written are correct?
            $endgroup$
            – Noor Aslam
            Jan 8 at 12:47










          • $begingroup$
            @NoorAslam They are correct
            $endgroup$
            – Shubham Johri
            Jan 8 at 18:01
















          1












          $begingroup$

          Notice that for the cardioid of revolution, $rhoge1$ for $0lephilepi/2$ and $rho<1$ for $pi/2<philepi$. It intersects the hemisphere only in the $xy$ plane. Therefore, $0lephilepi/2$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Sir the limits on the $rho$ and $theta$ I have written are correct?
            $endgroup$
            – Noor Aslam
            Jan 8 at 12:47










          • $begingroup$
            @NoorAslam They are correct
            $endgroup$
            – Shubham Johri
            Jan 8 at 18:01














          1












          1








          1





          $begingroup$

          Notice that for the cardioid of revolution, $rhoge1$ for $0lephilepi/2$ and $rho<1$ for $pi/2<philepi$. It intersects the hemisphere only in the $xy$ plane. Therefore, $0lephilepi/2$.






          share|cite|improve this answer









          $endgroup$



          Notice that for the cardioid of revolution, $rhoge1$ for $0lephilepi/2$ and $rho<1$ for $pi/2<philepi$. It intersects the hemisphere only in the $xy$ plane. Therefore, $0lephilepi/2$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 7 at 18:16









          Shubham JohriShubham Johri

          4,885717




          4,885717












          • $begingroup$
            Sir the limits on the $rho$ and $theta$ I have written are correct?
            $endgroup$
            – Noor Aslam
            Jan 8 at 12:47










          • $begingroup$
            @NoorAslam They are correct
            $endgroup$
            – Shubham Johri
            Jan 8 at 18:01


















          • $begingroup$
            Sir the limits on the $rho$ and $theta$ I have written are correct?
            $endgroup$
            – Noor Aslam
            Jan 8 at 12:47










          • $begingroup$
            @NoorAslam They are correct
            $endgroup$
            – Shubham Johri
            Jan 8 at 18:01
















          $begingroup$
          Sir the limits on the $rho$ and $theta$ I have written are correct?
          $endgroup$
          – Noor Aslam
          Jan 8 at 12:47




          $begingroup$
          Sir the limits on the $rho$ and $theta$ I have written are correct?
          $endgroup$
          – Noor Aslam
          Jan 8 at 12:47












          $begingroup$
          @NoorAslam They are correct
          $endgroup$
          – Shubham Johri
          Jan 8 at 18:01




          $begingroup$
          @NoorAslam They are correct
          $endgroup$
          – Shubham Johri
          Jan 8 at 18:01


















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