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Lecture

# 6.3 Volumes by Cylindrical Shells Question #2 (Easy)

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University of Toronto St. George

Mathematics

MAT136H1

all

Winter

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6.3 Integral Applications
Cylindrical Shells
Question #2 (Medium): Volume by Cylindrical Shells Rotated About the X-Axis
Strategy
First helpful step is to graph the given function and lines.
Recalling the volume calculation using cylindrical shells: ∫
Given function defines the interval[ ]for integral and which curve denotes the of the
cylindrical shells. If this function is given in the form of , when the rotation is about the -axis,
then rearrange the function to get the inverse . Then the volume is written as:
∫ .
If the function describing the height is given in the form of , no rearrangement is necessary,
and so it can be used directly for the integral.
Sample Problem
Using cylindrical shells find the volume obtained by rotating the region bound by the curves about the -
axis.
√ , ,
Solution
By graphing the function and the lines, i

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