Math 152 – Spring 2016
Section 7.3
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Section 7.3– Volumes by Cylindrical Shells
Find the volume of the solid created by rotationg the area enclosed between y = x2 − x3
and y = 0 around the y-axis.
Instead, we can use the method of cylindrical shells:
Theorem. The volume of a solid where the area under y = f (x) between x = a and
x = b is rotated around the y-axis is
Z
V =
b
2πxf (x) dx
a
Example 1. Find the volume of the following solids.
(a) The solid created by rotating the area enclosed between y = x2 − x3 and y = 0
around the y-axis.
Math 152 – Spring 2016
Section 7.3
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(b) The solid created by rotating the region bounded by x = y 2 , x = 0, y = 2, and
y = 5 around the x-axis.
(c) The solid created by rotation the region bounded by y 2 = x and x = 2y around the
y-axis.
Question. What if the area is not rotated around an axis?
Solution: The volume of a solid using cylindrical shells is
Z
b
circumf erence ∗ height dx
V =
a
where formulas are inserted for circumference (2πr) and height of each shell, and a and
b are the x-values where the shells start and stop.
Math 152 – Spring 2016
Section 7.3
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Example 2. Use cylindrical shells to find the volume generated by rotating the region
bounded by the curves around the specified line.
(a) y = x3 , y = 0, x = 1, x = 2; about the line x = 3
(b) x = 4 − y 2 , x = 2 − y; about the line y = 2
Math 152 – Spring 2016
Section 7.3
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Question. Should I use shells or disks/washers?
Remark. Be careful with your axis of rotation and your variable for integration:
Axis of rotation
Disks/Washers
Shells
x-axis or y = a
x integral
y integral
y-axis or x = a
y integral
x integral
Example 3. Graph both a typical disk/washer and typical shell for the following solids
and then setup the integrals. Which method is easier?
(a) Rotate the area bounded by the curves y = x2 , x = 1, x = 2, and y = 0 around the
y-axis.
(b) Rotate the area bounded by the curves y = ex , x = 1, and y = 0 around the x-axis.
(c) Rotate the area bounded by y = x2 and y = 2x around the y-axis.