How to find the volume by using shell methodI would like to find the volume of x 2 9 + y 2 = 1 rotated around the x axis, but using the shell method. How can I do this?Letting the thickness be dy and the problem I encountered is finding the radius.

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How to find the volume by using shell methodI would like to find the volume of             x      2        9    +      y    2    =  1 rotated around the x axis, but using the shell method. How can I do this?Letting the thickness be dy and the problem I encountered is finding the radius.

Clarissa Adkins · Accepted Answer

Step 1Disc method: Your ellipse gives   y  =  ±      1    −                  x        2            9      Since your are rotating about the x-axis, this gives a radius of       1    −                  x        2            9      , a disc circular area of   π      r    2   and so a volume of       ∫          −      3        3    π      (    1    −                  x        2            9        )    d  x.Step 2Shell method: Your ellipse gives   x  =  ±  3      1    −          y      2      Since your are rotating about the x-axis, this gives a cylindrical &quot;height&quot; of   6      1    −          y      2       and radius of y, a cylindrical shell surface area of   2  π  r  h and so a volume of       ∫          0        1    12  π  y      1    −          y      2          d  yIf you do the integrations you get the same answer. You would also get the same answer taking a sphere of radius 1 and then stretching it by a factor of 3 along the x-axis to give your ellipsoid

Find the volume of x^2/9+y^2=1 rotated around the x axis, but using the shell method. Let's set the thickness dy and find the radius.

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