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Dr. Lou Talman

Dept. of Mathematical
and Computer Sciences

Metropolitan State University

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Application of the Definite Integral

  Area of a Surface of Revolution

NCB Deposit  # 37

Surface area image

Surface Area 

Revolving a curve about an axis generates a surface area.

Definition:  If a function  y = f(x) has a continuous first derivative throughout the interval a < x < b, then the area of the surface generated by revolving the curve about the x-axis is the number


Replay the animation
More equations
Unfortunately, the integrals involving surface area are often quite tricky.

Another example . . .

Another animation
Replay the animation
This animation demonstrates how a surface area is generated by revolution and then how the sum of disks results in a volume.

Sum of disks results in volume

but the surface area is more complicated.
Surface area equations
This "SA" for the surface on the left
was calculated using Mathematica®.

Revolution about the  y- axis:
y-axis equation
y-axis image   More equations

Background for the student. . . .

Significance of Volumes and Surfaces

The definite integral is an amazingly versatile tool.  In Deposit #36 we see how a rotated plane figure sweeps out a volume.  In this Deposit #37 we see how a curve rotated about an axis sweeps out a surface with area.

A Famous Paradox  

Gabriel's Horn or Torricelli's Trumpet

Gabriel's Horn    

If the function  y = 1/x   is revolved around the x-axis for x > 1,
the figure has a finite volume, but infinite surface area.


Gabriel's Horn or Torricelli's Trumpet


Applications . . . .

Surface Area of a Solar Collector

Solar collector
Barstow, CA

Other applications
  • Satellite dish
  • Telescopes
  • Lenses
  • CAT scans
  • MRIs
  • Industrial designs
  • Construction,
e.g., Disney Hall in Los Angeles.

Disney Hall Los Angeles

Printed References
Modern calculus texts will have extensive material on area of a surface of revolution in the chapter on definite integrals.

James Stewart,  Calculus, 5th ed., THOMSONBrooks/Cole, 2003,  p. 590.

Howard Anton,  Calculus, 6th ed., John Wiley and Sons, 1999,  p. 485.

Smith and Minton,  Calculus, 2nd ed., McGraw-Hill, 2002,  p. 437.

For Mathematica® code that will create many variations of these graphs see
Gray, A.,  MODERN DIFFERENTIAL GEOMETRY of Curves and Surfaces with Mathematica®,  2nd. ed., CRC Press, 1998.

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