Determining the Surface Area of a Solid of Revolution

Key Questions

• How do you find the surface area of the solid obtained by rotating about the #y#-axis the region bounded by #y=x^2# on the interval #1<=x<=2# ?

  First of all, you are missing a bound. We will assume that the other bound is #y=0# or the #x#-axis. The answer is #(15pi)/2#.

  The first step is to determine whether you are rotating along an axis that is parallel to the independent axis or the axis of the parameter (#x# in this case). And we are not, so this integration should be done with cylindrical shells.

  
  Always draw a diagram to verify what is the parameter and what is the function.

  You should note that #y# is not always a parameter of #x#. For instance, #x=y^2#, #x# is now a parameter of #y#.

  The formula for cylindrical shells is:

  #V=int_a^b2pirhdr#
  #h# is represented by #y#, we have #y=x^2# and #y=0#
  #r# is represented by #x#
  #V=int_1^2 2 pi x (x^2-0) dx#

  Now that the substitutions are done, we can solve:

  #V=2 pi int_1^2x^3dx#
  #=2pi (x^4)/4|_1^2#
  #=2pi([2^4-1^4])/4#
  #=(15pi)/2#

  Amory W. · · Sep 5 2014
• How do you find the surface area of the solid obtained by rotating about the #x#-axis the region bounded by #y=e^x# on the interval #0<=x<=1# ?

  The answer is #pi/2[e^2-1]#.

  Since you are only given a single function and we are rotating about the axis of the parameter, this requires the disk method. The disk method is:

  #V=int_a^b Adx#
  #=int_a^b pi r^2dx#
  #=int_a^b pi [f(x)]^2dx#

  We have the known values:

  #f(x)=e^x#
  #a=0#
  #b=1#

  And now we can substitute:

  #V=int_0^1 pi (e^x)^2dx#
  #=pi int_0^1 e^(2x)dx#
  #=pi (e^(2x))/2|_0^1#
  #=pi/2[e^2-e^0]#
  #=pi/2[e^2-1]#

  Amory W. · · Sep 16 2014
• How do you find the surface area of a solid of revolution?

  If the solid is obtained by rotating the graph of #y=f(x)# from #x=a# to #x=b#, then the surface area #S# can be found by the integral

  #S=2pi int_a^b f(x)sqrt{1+[f'(x)]^2}dx#

  Wataru · · Sep 21 2014

• How do you find the surface area of a solid of revolution?
• How do you find the surface area of the solid obtained by rotating about the #y#-axis the region bounded by #y=x^2# on the interval #1<=x<=2# ?
• How do you find the surface area of the solid obtained by rotating about the #x#-axis the region bounded by #y=e^x# on the interval #0<=x<=1# ?
• How do you find the surface area of the solid obtained by rotating about the #x#-axis the region bounded by #x=1+2y^2# on the interval #1<=y<=2# ?
• How do you find the surface area of the solid obtained by rotating about the #y#-axis the region bounded by #y=1-x^2# on the interval #0<=x<=1# ?
• How do you find the surface area of the solid obtained by rotating about the #x#-axis the region bounded by #9x=y^2+18# on the interval #2<=x<=6# ?
• How do you find the surface area of the solid obtained by rotating about the #x#-axis the region bounded by #y=x^3/6+1/(2x)# on the interval #1/2<=x<=1# ?
• How do you find the surface area of the part of the circular paraboloid #z=x^2+y^2# that lies inside the cylinder #x^2+y^2=1#?
• How do you determine the surface area of a solid revolved about the x-axis?
• How do you find the centroid of the quarter circle of radius 1 with center at the origin lying in the first quadrant?
• What is the surface area produced by rotating #f(x)=1-x, x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=abs(1-x), x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=x^3-x^2+1, x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=sinx-cosx, x in [0,pi/4]# around the x-axis?
• What is the surface area produced by rotating #f(x)=sinx/cosx, x in [0,pi/4]# around the x-axis?
• What is the surface area produced by rotating #f(x)=tanx-cos^2x, x in [0,pi/4]# around the x-axis?
• What is the surface area produced by rotating #f(x)=cscxcotx, x in [pi/8,pi/4]# around the x-axis?
• What is the surface area produced by rotating #f(x)=x^2lnx, x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=2x^2+4x+3, x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=2/x-1/x^2, x in [1,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=1/x-1/(x+3), x in [1,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=1/(x^2+1), x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=1/(x+1), x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=(1-x)/(x^2+6x+9), x in [0,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=x^3-8, x in [0,2]# around the x-axis?
• What is the surface area produced by rotating #f(x)=2/(e^x-3), x in [0,2]# around the x-axis?
• What is the surface area produced by rotating #f(x)=3x^3+6x^2-2x+3, x in [-3,2]# around the x-axis?
• What is the surface area produced by rotating #f(x)=x/pi^2, x in [-3,3]# around the x-axis?
• What is the surface area produced by rotating #f(x)=e^(x^2), x in [-1,1]# around the x-axis?
• What is the surface area produced by rotating #f(x)=1/e^(x^2), x in [-1,1]# around the x-axis?
• What is the surface area produced by rotating #f(x)= xtan2x -tanx , x in [pi/12,(11pi)/12]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=x^2# for #x in [1,2]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=1/x^2# for #x in [1,2]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=x-1# for #x in [1,2]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=(x-3/2)^2# for #x in [1,2]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=sqrt(x^3)# for #x in [1,2]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=sqrt(x)# for #x in [1,2]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=sqrt(4x)# for #x in [0,1]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=xsqrt(x+1)# for #x in [0,1]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=xsqrt(x-1)# for #x in [2,3]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=2-x# over #x in [2,3]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=-2x^3-3x^2+6x-12# over #x in [2,3]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=-sqrtx/e^x# over #x in [0,1]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=e^(x^2-x)/(x+1)-e^x# over #x in [0,1]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=e^(x^2+x-1)/(x+1)# over #x in [0,1]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x)=e^(x+1)/(x+1)# over #x in [0,1]# around the x-axis?
• What is the surface area of the solid created by revolving #f(x) = 1/x, x in [2,5]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 1/(1-x), x in [2,5]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 3x, x in [2,5]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = x^2-x , x in [2,7]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = e^(x)/2 , x in [2,7]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = e^(x-2) , x in [2,7]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = e^(2x) , x in [2,7]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = xe^(2x) , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = xe^-x-xe^(x) , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = xe^-x-e^(x) , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = e^-x+e^(x) , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = (2x-2)^2 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = (3x-2)^2 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = (3x-1)^2 , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = (2x-9)^2 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = (x-9)^2 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = lnx , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 1/(x+e^x) , x in [3,4]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = x^2-3x+2 , x in [3,4]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 2x^2-x+5 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 2x^2-4x+8 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 3x^2-4x+8 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 2x^2-6x+18 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = x^2-2x+15 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 2x^2-5x+15 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 6x^2-3x+22 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 2x^2-3x+22 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = x^2-3x+24 , x in [2,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = x^2+e^x , x in [2,4]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = 2x^2+3 , x in [1,4]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) = x^3 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =2x+5 , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =e^(3x-2) , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =e^(4x)-e^(2x) , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =e^(4x)-x^2e^(2x) , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =e^(2-x) , x in [1,2]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =e^(2-4x^2) , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =ln(2x) , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =ln(4-x) , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =x^2ln(x)-x , x in [1,3]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =2x^3-2 , x in [2,4]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =2x^3-2x , x in [2,4]# around the x axis?
• What is the surface area of the solid created by revolving #f(x) =x^3+6x , x in [2,5]# around the x axis?
• How do you find the exact area of the surface obtained rotating the curve about the #x#-axis of #y=sqrt(8-x)#, #2<=x<=8#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #y=x^2, 0<=x<=2#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #y=cx+d, a<=x<=b#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #y=1/3(x^2+2)^(3/2), 1<=x<=2#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #y=1/4x^4+1/8x^-2, 1<=x<=2#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #x=2t+1, y=4-t, 0<=t<=4#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #x=t+1, y=1/2t^2+t, 0<=t<=2#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #x=t^2, y=1/3t^3, 0<=t<=3#?
• How do you find the area of the surface generated by rotating the curve about the y-axis #x^(2/3)+y^(2/3)=1# for the first quadrant?
• How do you find the area of the surface generated by rotating the curve about the x-axis #y=1/3x^3, 0<=x<=1#?
• How do you find the area of the surface generated by rotating the curve about the x-axis #y=1/4x^4+1/8x^-2, 1<=x<=2#?
• How do you find the area of the surface generated by rotating the curve about the x-axis #x=t^2+t, y=2t+1, 0<=t<=1#?
• Question #a888b
• a) Show that the formula for the surface area of a sphere with radius #r# is #4pir^2#. b) If a portion of the sphere is removed to form a spherical cap of height #h# then then show the curved surface area is #2pihr^2#?
• Question #678c2
• What is the area of the region in the first quadrant bounded on the left by the y-axis, below by the line y = x/4, above left by the curve y=1 + sqrt(x), and above right by the curve y=2\/sqrt(x)?