6.3

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6.3 Volumes of Cylindrical Shells
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Derivation
Assume you have a functions similar to the one
shown below and assume the f is to difficult to
solve for y in terms of x. Rotate f about the y
axis.
It is too difficult to find xi and xo so that we
can find the area of the washer. We need another
method for more complicated functions.
y f (x)
xi
xo
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Cylindrical Shells
Let xi be some subinterval and establish a
rectangle to estimate the area. When this
rectangle is rotated about the y-axis, a
cylindrical shell if formed.
y f (x)
xi
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Cylindrical Shells
Now determine the volume of the cylindrical
shell. V2 is the volume of the larger right
circular cylinder and V1 is the volume of the
inner.
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Cylindrical Shells
Now assume that the interval a, b is divided
into n subintervals of equal width ?x. Also
assume that xi is taken at the mid-point of each
subinterval. This means that ri xi the height
of each cylindrical shell is given by f (xi). The
volume of each cylindrical shell is
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Cylindrical Shells
An estimate of the total volume is the sum of the
volumes of the n cylindrical shells.
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Volume of a Solid Using Cylindrical Shells
Therefore, the volume of the solid obtained by
rotating about the y-axis the region under the
curve y f (x) from a to b is
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Volume of a Solid Using Cylindrical Shells
Assume you have a functions similar to the one
shown below and assume the f is to difficult to
solve for y in terms of x. Rotate f about the y
axis.