Back to . . . . Dr. Lou Talman Dept. of Mathematical and Computer Sciences Metropolitan State University of Denver Application of the Definite Integral   Area of a Surface of Revolution NCB Deposit  # 37 Calculus of 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

Unfortunately, the integrals involving surface area are often quite tricky.

 Another example . . .

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.

but the surface area is more complicated.

This "SA" for the surface on the left
was calculated using Mathematica®.

For more of Dr. Talman's animations see
http://rowdy.msudenver.edu/%7Etalmanl/APCalculus.html

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.

Gabriel's Horn or Torricelli's Trumpet

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

Barstow, CA

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