6.2 Volumes of Revolution (Part III) - Washer Method
The washer method is essentially but two applications of the disk method. Suppose that
f(x) > g(x) > 0,
for all x in the interval I =[a , b].
Further assume that f and g are bounded over that interval.
We'll find the volume of the solid of revolution that results when the region bounded by the graphs of f and g over interval I (the upper yellow region in the diagram) revolves about the x-axis.At a point x on the x-axis, a perpendicular cross section of the solid consists of the region between two concentric circles (shaped like a washer, it's called an annulus). The outer circle has radius R = f(x), and the inner circle (the hole) has radius r = g(x).
The area of an annulus is just the area of the big (outer) circle minus the area of the smaller (inner) circle: A = pR^{2 }– pr^{2 }= p(R^{2 }– r^{2})
Thus the volume of a thin annular slice (washer) of width dx is
V_{washer} = p(R^{2 }– r^{2})dx.
Now we need but add up all these washer volumes, from x=a to x=b:
This animation shows the solid obtained when the
region depicted in the previous diagram revolves
about the x-axis.
The x-axis can be seen running through the middle of the solid. |
Since
p(R^{2 }– r^{2})dx = pR^{2}dx^{ }– pr^{2}dx,
the washer method is equivalent to applying the disk method twice - once for the overall solid, and another to compute the volume of the hollowed out cavity within the overall solid. The cavity volume is then subtracted from the volume that enfolds it, to obtain the volume of solid of revolution.
Example 1: Find the volume of the solid of revolution formed by rotating the finite region bounded by the graphs of about the x-axis.
This is the same region featured in Example 2 in the shell method section, but this time we're revolving it about the x-axis instead of the y-axis. The x-axis picture is blue (from x=0 to x=4) in this rotating depiction of the solid of revolution.
Example 2: Find the volume of the solid of revolution formed by rotating the region bounded by the graphs of over the interval [1, p], about the x-axis.
6.2 Exercises Do the problems that can be solved with the washer method.
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