Determining the Surface Area of a Solid of Revolution

Key Questions

• How do I find the surface area of a solid of revolution using parametric equations?

  If a surface is obtained by rotating about the x-axis from #t=a# to #b# the curve of the parametric equation

  #{(x=x(t)),(y=y(t)):}#,

  then its surface area A can be found by

  #A=2pi int_a^by(t)sqrt{x'(t)+y'(t)}dt#

  If the same curve is rotated about the y-axis, then

  #A=2pi int_a^b x(t)sqrt{x'(t)+y'(t)}dt#

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  I hope that this was helpful.

  Wataru · · Oct 18 2014

• What is the surface area of the solid created by revolving #f(t) = ( 3t-1, t^2-2t+2), t in [2,3]# around the x-axis?
• What is the surface area of the solid created by revolving #f(t) = ( t^3-3t+4, t^3-2t, t in [2,3]# around the x-axis?
• What is the surface area of the solid created by revolving #f(t) = ( t^3-t, t^3-t, t in [2,3]# around the x-axis?
• Question #99004
• How do you find the surface area generated when the curve #y=sqrt(r^2-x^2), 0<=x<=a# is rotated about the x axis?