Determining the Surface Area of a Solid of Revolution

Key Questions

• How do I find the surface area of the solid defined by revolving #r = 3sin(theta)# about the polar axis?

  Write :

  #r = 3 sin(theta)#

  #r = 3y/r# because #y=r sin(t)#

  #r^2 = 3y#

  #x^2 + y^2 = 3y# because #r = sqrt(x^2+y^2)#.

  #x^2 + (y-3/2)^2 = 9/4#

  You recognize a circle of radius #3/2#. The area is #pi (3/2)^2 = 9 pi/ 4#.

  vince · 1 · Feb 24 2015
• How do I find the surface area of a solid of revolution using polar coordinates?

  If a surface is obtained by rotating about the x-axis the polar curve #r=r(theta)# from #theta=theta_1# to #theta_2#, then its surface area A can by found by

  #A=2pi int_{theta_1}^{theta_2}r(theta)sin theta sqrt{r^2+[r'(theta)]^2} d theta#.

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  I hope that this was helpful.

  Wataru · · Oct 18 2014