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Dr. Lou Talman
Metro State

Dept. of Mathematical
and Computer Sciences

 http://clem.mscd.edu/~talmanl


Application of the Definite Integral

Volume of a Solid of Revolution

NCB Deposit  # 36

Calculus
of
Volumes 
Revolving a plane figure about an axis generates a volume.

Definition:  Consider the region between the graph of a continuous function  y = f(x) and the x-axis from  x =  a to x =  b.



 

Plane region
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Another example . . .



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The function need not be one of the standard plane figures found in elementary geometry.
The perpendicular cross-section, or slice, is still a circle.



Another version . . .


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Revolution about the  y- axis:
For more of Dr. Talman's animations see
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Note:  If  the cross-section is NOT a disk, but a washer, we first write the area of the washer by subtracting the area of the inner cross-section from the area of the outer cross-section.   Then we set up the integral,  being careful to choose whether the rotation is about x-  or  y-.


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.
But the process of visualizing this one concept has far wider applications.


We can compute area, volume, arc length and surface area using essentially the same mental process. First we divide an object into smaller pieces -  n smaller pieces of a thickness that will eventually become our dx or dy.  We approximate a quantity for each of the small pieces.  This is usually an area or a length.  We add up the approximations and then take a limit.  Thus, we have intuitively derived a definite integral.
  • Sketch the solid and a typical cross section.
  • Find a formula for the crioss-sectional area A(x).
  • Find the limits for integration on the rotational axis.
  • Integrate A(x) to find the volume.

A Famous Paradox  

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




Another example:



Note that  y in the equation has only the first power and becomes the axis of rotation for this elliptical paraboloid.

Volume of a Circular Paraboloid

Printed References
Modern calculus texts will have extensive material on volume of solids of revolution in the chapter on definite integrals.

James Stewart,  Calculus, 5th ed., THOMSONBrooks/Cole, 2003,  p. 382.

Howard Anton,  Calculus, 6th ed., John Wiley and Sons, 1999,  p. 468.

Finney, Weir, and Giordano, Thomas' Calculus, 10th ed., Addison-Wesley, 2001,  p. 415.

Smith and Minton,  Calculus, 2nd ed., McGraw-Hill, 2002,  p. 411.

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.
Applications
  • CAT scans
  • MRIs
  • Industrial designs
  • Containers and packaging
  • Construction
  • Weight of a part turned on a lathe.