Calculation
of the Volume of Revolution using the Method of Shells
|
In a Nut Shell: Calculation of the volume generated
by rotating a curve or a set of curves about a designated axis of
rotation is a three step process such as: For the Method of Shells Step 1: Identify the element of
volume, dV, and show it on the graph of y(x) For a shell element dV = 2πr
w dy. See figure below. Step 2: Determine the limits of integration
for the region (volume to be calculated) Step 3: Evaluate the integral |
|
Step 1
Note that the shell is generated by rotating the hatched area shown about
the axis of rotation. |
|
Steps 2 and 3 Establish limits of
integration. In this case the
integration is from y = f(a) to y
= f(b). Express the total volume
as an integral over the region rotated about the axis of rotation. The disk is defined by dV = [ 2π r
w] dy
where r = f(x) + h and w = b
– x =
b – g(y) y = f(b) V
= ∫ [ 2π ( f(x) + h
) ( b – x) ] dy where
x = g(y) y = f(a) |
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