What is the surface area produced by rotating f ( x ) = e x 2 , x ∈ [ − 1 , 1 ] around the x-axis?

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What is the surface area produced by rotating   f  (  x  )  =      e                  x                  2                      ,  x  ∈  [  −  1  ,  1  ] around the x-axis?

hopeloothab9m · Accepted Answer

What you&#039;d do to determine the surface area for a solid of revolution around the x axis is take the equation for the circumference and integrate over the surface.  S  =  2  π      ∫          −      1              1        r  (  x  )  d  S  (  x  )As it turns out, the differential surface can be treated as the arc length for an infinitesimal distance along the chosen axis. So:  d  S  (  x  )  =      1    +    (          r      ′        (    x    )          )              2              d  x,and:  S  =  2  π      ∫          −      1              1        r  (  x  )      1    +    (          r      ′        (    x    )          )              2              d  xFirst, let&#039;s evaluate the squared derivative:      d          d      x        [      e                  x                  2                      ]  =  2  x      e                  x                  2                      (  r  (  x  )      )          2        =  4      x          2            e          2              x                  2                    This gives:  S  =  2  π      ∫          −      1              1            e                  x                  2                          1    +    4          x              2                    e              2                  x                      2                                d  x  =  2  π      ∫          −      1              1            e          2              x                  2                                e              −        2                  x                      2                                +    4          x              2              d  xThere is however, no result in terms of elementary functions. Numerically, this integral is 46.3958

What is the surface area produced by rotating f ( x ) = e <mrow class="MJX-TeX

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