volumes of solids of revolution. questions involving the area of a region between curves, and the...
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VOLUMES OF SOLIDS OF REVOLUTION
• Questions involving the area of a region between curves, and the volume of a solid formed when this region is rotated about a horizontal or vertical line, appear regularly on both the AP Calculus AB and BC exams.
• Students have difficulty when the solid is formed by use a line of rotation other than the x- or y-axis.
• These types of volume are part of the type of volume problems students must solve on the AP test.
• Students should find the volume of a solid with a known cross section.
• The Shell method is not part of the AB or the BC course of study anymore.
• The four examples in the Curriculum Module use the disk method or the washer method.
Example 1 Line of Rotation Below the Region to be Rotated
• Picture the solid (with a hole) generated when the region bounded by and are revolved about the line y = -2.
• First find the described region• Then create the reflection
over the line y=-2
2y x xy e -1 1
-6
-5
-4
-3
-2
-1
1
Example 1
• Think about each of the lines spinning and creating the solid.
• Draw one representative disk.
• Draw in the radius.
Example 1• Find the radius of the
larger circle, its area and the volume of the disk. 2 outsider y
22 outsideArea y
2
2
2
2 2
outsideVolume y x
x x
• Sum up these cylinders to find the total volume
21
2 2n
k
Volume x x
The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.
• The points of intersection can be found using the calculator.
• Store these in the graphing calculator
(A=-1.980974,B=0.13793483)(C=0.44754216,D=1.5644623)
• Write an integral to find the volume of the solid.
22 2 71.9833363C
A
Volume x dx
Example 1• Find the radius of the
smaller circle, its area and the volume of the disk. 2 insider y
22 insideArea y
2
2
2
2
inside
x
Volume y x
e x
• Sum up these cylinders to find the total volume
21
2n
x
k
Volume e x
The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.
• Using the points of intersection write a second integral for the inside volume.
(A=-1.980974,B=0.13793483)(C=0.44754216,D=1.5644623)
22 52.258610C
x
A
Volume e dx
Example 1• The final volume will
be the difference between the two volumes.
2 22 2 2 19.724 19.725
C Cx
A A
x dx e dx or
Example 2 Line of Rotation Above the Region to be Rotated
• Rotate the same region about y = 2
• Notice that 2y x xy e
2
2outside
outside
y radius
r y
Example 2 Line of Rotation Above the Region to be Rotated
• The area of the larger circle is
2
2
2
2
outside
x
y
e
22 xVolume e x
Example 2 Line of Rotation Above the Region to be Rotated
• The sum of the volumes is
2
1
2
2
2
16.406065
nx
k
Cx
A
Volume e x
e dx
Example 2 Line of Rotation Above the Region to be Rotated
• The area of the smaller circle is
2
2
2
2 2
insidey
x
2
2 2Volume x x
Example 2 Line of Rotation Above the Region to be Rotated
• The sum of the volumes is
.
2
1
2
2 2
2 2
7 870360
Vn
k
C
A
Volume x x
x dx
Example 2 Line of Rotation Above the Region to be Rotated
• The volume of the solid is the difference between the two volumes
222 2 2
8.535 8.536
C C
A A
Volume
e dx x dx
or
Example 3 Line of Rotation to the Left of the Region to be Rotated
• Line of Rotation: x = -3• Use the same two
functions• Create the reflection• Draw the two disks and
mark the radius
Example 3 Line of Rotation to the Left of the Region to be Rotated
• The radius will be an x-distance so we will have to write the radius as a function of y.
2y x xy e
2
2
2
2
y x
x y
ln
xy e
y x
Example 3 Line of Rotation to the Left of the Region to be Rotated
The radius of the larger disk is 3 + the distance from the y-axis or 3 + (ln y) Area of the larger circle is
23 lny
Example 3 Line of Rotation to the Left of the Region to be Rotated
• Volume of each disk:
23 lnVolume y y
23 lnD
BVolume y dy
Example 3 Line of Rotation to the Left of the Region to be Rotated
• The radius of the smaller disk is
• 3+ the distance from the y-axis or 3 + (y2 – 2)
• Area of the larger circle is
223 ( 2)y
Example 3 Line of Rotation to the Left of the Region to be Rotated
• Volume of each disk:
223 ( 2)Volume y y
223 ( 2)D
BVolume y dy
Example 3 Line of Rotation to the Left of the Region to be Rotated
• Difference in the volume is
22 23 (ln ) 3 ( 2)
15.538 15.539
D D
B By dy y dy
or
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Line of Rotation: x = 1
• Create the region, reflect the region and draw the disks and the radius
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Notice the larger radius is 1 + the distance from the y-axis to the outside curve.
• The distance is from the y-axis is negative so the radius is
21 ( 2)y
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Area of Larger disk:
• The volume of the disk is
221 ( 2)y
221 2 y x
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Volume of all the disks are
22
1
22
1 ( 2)
1 ( 2)
n
k
D
B
y y
y dy
V
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Area of smaller disk:
• The volume of the disk is
21 (ln )y
21 lny y V
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Volume of all the disks are
2
1
2
1 ln
1 ln
n
k
D
B
y y
y dy
V
Example 4 Line of Rotation to the Right of the Region to Be Rotated
• Find the difference in the volumes
2 221 ( 2) 1 ln 12.72067D D
B By dy y dy