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Applications of the Integral

Volumes of Solids

Problems

Volumes of Solids, page 2

page 1 of 2

The application of integrals to the computation of areas in the plane can be extended to the computation of certain volumes in space, namely those of solids of revolution. A solid of revolution arises from revolving the region below the graph of a function f (x) about the x - or y -axis of the plane. A cone arises in this way from a triangular region, a sphere from a semicircular region, and a cylinder from a rectangular region. These are just a few of the possibilities for solids of revolution.

There are two primary methods for finding the volume of a solid of revolution. The shell method is applied to a solid obtained by revolving the region below the graph of a function f (x) from a to b about the y -axis. It approximates the solid with a number of thin cylindrical shells, obtained by revolving about the y -axis the thin rectangular regions used to approximate the corresponding region in the plane. This is illustrated in the figure below.

Figure %: The Shell Method of Finding the Volume of a Solid of Revolution

The volume of a thin cylindrical shell of radius x , thickness Δx , and height f (x) is equal to


Π(x + )2 f (x) - Π(x - )2 f (x) = Π(2xΔx)f (x)  
  = (2Πx)(Δxf (x))  

Here by "cylindrical shell" we mean the region between two concentric cylinders whose radii differ only very slightly; precisely speaking, this formula is not correct for any positive thickness, but approaches the correct value as the thickness Δx shrinks to zero. Since we will ultimately consider such a limit, this formula will yield the correct volume in our application.

If we sum together the volumes of a family of such cylindrical shells, covering the entire interval from a to b , and take the limit as Δx→ 0 (and consequently as the number of cylindrical shells approaches infinity), we end up with the integral

Vol = 2Πxf (x)dx = 2Π xf (x)dx    

The disk method for finding volumes applies to a solid obtained by revolving the region below the graph of a function f (x) from a to b about the x -axis. Here the solid is approximated by a number of very thin disks, standing sideways with the x -axis through their centers. These disks are obtained by revolving about the x -axis the thin rectangular regions used to approximate the area of the corresponding region in the plane. This is illustrated in the figure below.

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