CHAPTER 16
Mrs.Volynskaya AP Calculus Study Guide Disc/Washer Method
Question: The region bounded by [pic], [pic], [pic], and the [pic] axis, is rotated
360 ° about the [pic]-axis. Find the volume of the figure so formed.
Answer:
[pic]
Consider a thin vertical strip width [pic], height [pic]. When it is rotated about the [pic]-axis it forms a disc whose volume is [pic].
The volume of the solid [pic]
[pic]
[pic] (approx.)
Washer Method
Question: The region bounded by [pic] and [pic] is rotated 360 ° about the
[pic]-axis. Find the volume of the solid so formed.
Answer: [pic]
[pic] intersects [pic] at (2,8) and (0,0). We wish to find the volume of the “crescent” shaped region rotated around the [pic]-axis.
Consider a thin vertical strip width [pic].
[pic]
When it is rotated around the [pic]-axis it becomes a “washer”
[pic]
whose volume is [pic] where [pic], [pic]
refer to the radii of the large and small circle respectively.
Note that [pic] and [pic].
Volume of the washer [pic]
[pic]
[pic] Volume of solid of revolution [pic]
[pic]
Example
In the previous example, if the region had been rotated about the [pic]-axis then its volume could have been found as follows.
[pic]
Consider a thin horizontal strip which, when rotated about the [pic]-axis, would again form a “washer” whose volume is [pic] where [pic], [pic] are the large and small radii respectively.
[pic]
[pic] is the [pic] co-ordinate of a point on [pic] and [pic] is the [pic] co-ordinate of a point on [pic]. i.e. [pic] and [pic].
The volume of the washer [pic]
The volume of the solid of revolution [pic]
[pic].
Example
Question: Consider the region [pic] bounded by [pic] and [pic]. Find the volume of the solid formed when this region [pic] is rotated about the
[pic]-axis.
Answer:
[pic]
Clearly it is impractical to use horizontal strips for values of [pic] greater than 2 and hence a vertical strip is used. The thin vertical strip, when rotated about the [pic]-axis produces a washer whose volume is [pic].
The volume of the solid of revolution:
[pic]
[pic]
[pic]
[pic] (approx.)
Example
Shell Method
Question: Consider the solid formed when the region [pic] in the last example is rotated about the [pic]-axis. Find the volume of the solid so formed.
Answer: As explained it is impractical to use a horizontal strip and hence the thin vertical strip, when rotated 360 ° about the [pic]-axis produces a shape known as a shell whose dimensions are shown in the figure below. The word shell is used in its military, rather than marine, sense.
[pic]
To evaluate the volume of the shell so formed, imagine cutting open the shell to produce a lamina as shown below
[pic]
whose dimensions are as shown.
The volume of the lamina is [pic].
The volume of the solid of revolution formed is
[pic]
[pic]
[pic]
[pic]
[pic]
Example
Question: The region bounded by the [pic]-axis and [pic] is rotated around the vertical line [pic]. Find the volume of the solid so formed.
Answer:
[pic]
When rotated the vertical thin strip becomes a solid as shown below.
[pic]
The volume of this thin shell considered as a lamina when cut open is
[pic]
[pic]
Total volume of solid of revolution
[pic]
[pic]
[pic]
[pic].
Worksheet 1
1. Find the volume of the solid of revolution formed by rotating the
ellipse [pic] a) about the [pic]-axis. b) about the [pic]-axis.
2. Find the volume of the solid generated by rotating about the [pic]-axis, the area formed by [pic], the [pic]-axis, and [pic].
3. Find the volume of the solid generated by rotating about the line [pic], the area bounded by [pic], the [pic]-axis, and [pic].
4. [pic] is the origin, [pic] is (2,0), [pic] is (2,2), and [pic] is (0,2).
a) Find the volume of the solid of revolution formed by rotating triangle
[pic] about the [pic]-axis.
b) Find the volume of the solid of revolution formed by rotating triangle
[pic] about the [pic]-axis.
c) Since clearly area of [pic] = area of [pic], why are the volumes
generated different?
5. Find the volume of the solid generated by rotating the region bounded by [pic] and [pic] a) about the [pic]-axis and b) about the line [pic].
6. The graph of [pic] is drawn below.
[pic]
Find the volume of the solid generated by rotating the loop around the [pic]-axis.
7. The area bounded by the curve [pic], the [pic]-axis, [pic], and [pic] is rotated about the [pic]-axis. Find the volume of the solid figure so formed.
Answers to Worksheet 1
1. a) [pic] b) [pic] 2. [pic] 3. [pic] 4. a) [pic] b) [pic]
5. a) [pic] b) [pic] 6. [pic] 7. [pic]
Worksheet 2 – Calculators Permitted
1. The region [pic] is bounded above by the graph of [pic], on the left by [pic], on the right by [pic], and below by [pic]. The volume of the solid of revolution formed when [pic] is revolved about the line [pic] is nearest in value to:
(A) 6.8 (B) 7.0 (C) 7.2 (D) 7.4 (E) 7.6
2. The region [pic] in the first quadrant is bounded by [pic] and [pic]. If [pic] is revolved about the [pic]-axis, the volume of the solid formed is nearest in value to:
(A) 8.16 (B) 8.26 (C) 8.36 (D) 8.38 (E) 8.42
3. Let [pic] be the region in the first quadrant bounded by the [pic]-axis, the line [pic] and the graph of [pic]. What is the volume of the solid generated by rotating [pic] about the [pic]-axis?
(A) 2.67 (B) 2.70 (C) 2.73 (D) 2.76 (E) 2.79
4. Let [pic] be the region in the first quadrant bounded by the [pic]-axis and the curve [pic]. The volume produced when [pic] is revolved about the [pic]-axis is:
(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]
5. Let [pic] be the region in the first quadrant bounded above by the graph of [pic] and below by the graph of [pic]. What is the volume of the solid generated when [pic] is rotated about the [pic]-axis?
(A) 1.21 (B) 2.68 (C) 4.17 (D) 6.66 (E) 7.15
6. Let [pic] be the region enclosed by the graphs of [pic], [pic], and the line [pic]. The volume of the solid generated when [pic] is revolved about the
[pic]-axis is nearest to
(A) 33.09 (B) 33.11 (C) 33.13 (D) 33.15 (E) 33.17
7. Let [pic] be the region in the first quadrant enclosed by the graphs of [pic] and the lines [pic] and [pic]. What is the volume of the solid generated when [pic] is rotated about the [pic]-axis?
(A) 15.9 (B) 18.7 (C) 40.1 (D) 50.6 (E) 64.9
8. Which definite integral represents the volume of a sphere with radius 2?
(A) [pic] (B) [pic] (C) [pic]
(D)[pic] (E) [pic]
Answers to Worksheet 2
1. A 2. D 3. B 4. A 5. E 6. B 7. C 8. C
Volumes of Solids with Known Cross-Section
It is important to remember that integration is the process of adding together an infinite number of small “things”. For example suppose we had a closed vessel whose cross-section area, parallel to the base, was [pic] square metres where [pic] represented the distance in metres of the cross-section region from the base.
The base area, when [pic], would be 36 square metres and the height of the vessel would be 6 metres because, when [pic] the cross-section area equals zero. The vessel would resemble a “beehive” as shown.
[pic]
If we wished to find the volume of the “beehive” then we would simply evaluate [pic] since [pic] represents the volume of a thin horizontal strip and the integration process adds them all up.
Volume [pic] cubic metres.
Similarly consider the following question taken from an Advanced Placement Calculus AB examination.
[pic]
Let [pic] be the region in the first quadrant enclosed by the graphs of [pic], [pic], and the [pic]-axis, as shown in the figure above.
a) Find the area of the region [pic].
b) Find the volume of the solid generated when the region [pic] is revolved about the
[pic]-axis.
c) The region [pic] is the base of a solid. For this solid, each cross section
perpendicular to the [pic]-axis is a square. Find the volume of this solid.
Answer: a) We need to find first the point of intersection [pic] of [pic] and
[pic]. By calculator, [pic] is (0.94194408, 0.41178305).
By considering a thin vertical strip it is clear that the area of region [pic] (approx.)
b) Similarly by considering a thin vertical strip, when rotated around
the [pic]-axis, it becomes a washer whose volume is
[pic].
The volume of the solid of revolution so formed is
[pic][pic] (approx.)
c) The volume of each square cross-section is [pic].
and hence the total volume of the solid is
[pic]
Example
Question: Shown below is the base of a solid represented by [pic]. The solid is formed by having cross-sections in planes perpendicular to the [pic]-axis between [pic] and [pic] and are isosceles right-angled triangles with one side in the base. Find the volume of the solid.
[pic]
Answer: Each cross section has width [pic] and area [pic].
[pic] each cross section has volume [pic].
The total volume is hence [pic][pic]
[pic][pic]
An interesting theorem concerning rotations of regions is Pappus’ Theorem which states that the volume of a solid formed by rotating a region [pic] about a line, not intersecting the region, equals the area of the region [pic] times the distance travelled by the centre of gravity of the region in the rotation.
For example, consider the volume of the solid formed by rotating the circle [pic] about the line [pic].
[pic]
The volume of the torus (doughnut) so formed
= (Area of circle) times (circumference of circle formed when (0,0) rotates
360 ° around [pic])
= [pic] times [pic]
= [pic]
= 3947.8 (approx.)
Worksheet 3
1. The area bounded by the [pic]-axis, the [pic]-axis, [pic] and [pic] is rotated about the [pic]-axis. Find the volume of the solid so formed.
2. A hemispherical bowl of radius 5 cm contains water to a depth of 3 cm. Find the volume of the water.
3. The area in the first quadrant bounded by [pic], [pic], and the [pic]-axis is rotated about the [pic]-axis. Find the volume of the solid generated.
4. Prove that the volume of a sphere is [pic] considering the rotation of the circle [pic] about the [pic]-axis.
5. A container is such that its cross-section area is [pic] cm2 where [pic] is the distance in cm from the base.
a) What is the height of the container?
b) What is its volume?
6. The area of a cross-section of a vase at a distance [pic] cm below the top is [pic] cm2. Find the depth of the water when the vase is half full.
7. Find the volume of the solid generated by rotating the area bounded by [pic] and [pic]
a) about the [pic]-axis.
b) about the [pic]-axis.
8. Find the volume generated when the region enclosed by the lines [pic], [pic], and [pic] between (1,1) and (3, [pic]) is rotated about the [pic]-axis.
9. A closed vessel tapers to point [pic] and [pic] at its ends and is such that its cross-section area cut by a plane perpendicular to [pic], [pic] cm from [pic], is
[pic] cm2. Find the volume of the vessel.
10. The area bounded by the [pic]-axis, [pic], and [pic] is rotated about the
[pic]-axis. Find the volume of the solid generated.
11. Find the volume formed by rotating the area bounded [pic], the [pic]-axis and [pic],
a) about the [pic]-axis.
b) about the line [pic].
12. A solid is 12 inches high. The cross-section of the solid at height [pic] above its base has area [pic] square inches. Find the volume of the solid.
13. A solid extends from [pic] to [pic]. The cross-section of the solid in the plane perpendicular to the [pic]-axis is a square of side [pic]. Find the volume of the solid.
14. A solid is 6 ft high. Its horizontal cross-section at height [pic] ft above the base is a rectangle with length [pic] ft and width [pic] ft. Find the volume of the solid.
15. A solid extends along the [pic] axis from [pic] to [pic]. Its cross-section at any point [pic] is an equilateral triangle with edge [pic]. Find the volume of the solid.
Answers to Worksheet 3
1. 48.59 2. [pic] 3. [pic] 4. ---- 5. a) 2 b) [pic]
6. 0.695 7. a) [pic] b) [pic] 8. [pic] 9. [pic]
10. [pic] 11. a) [pic] b) [pic] 12. 216 13. [pic]
14. 132 15. [pic]
Worksheet 4
1. The finite area in the first quadrant bounded by the curves [pic], [pic] and the line [pic] is rotated once about the [pic]-axis. Find the volume of the solid formed.
2. Find the area enclosed by [pic], [pic] and the [pic]-axis.
3. Find the volume of the solid formed by rotating the circle [pic] about the [pic]-axis. (Pappus’ Theorem required)
4. The area bounded by the [pic]-axis, [pic] and [pic] is rotated about the line [pic]. Find the volume of the solid generated.
5. Find the volume of the solid generated by rotating about the [pic]-axis, the region, in the first quadrant, bounded by [pic] and [pic].
6. Find the volume of the solid formed by rotating the area bounded by [pic], the [pic]-axis and [pic] about the line [pic].
7. Find the volume of the solid formed by rotating [pic] about its major axis.
8. The region in the first quadrant bounded by [pic] is rotated about the [pic] Find the volume of the solid formed.
9. A hole of radius [pic] is drilled through a solid sphere of radius [pic], with one edge of the hole passing through the centre of the sphere. The volume of the material removed is [pic] where [pic] is integer. Find the value of [pic].
10. Find the volume of the solid formed by rotating [pic] around the [pic]-axis. (This means the finite area above the [pic]-axis between [pic] and [pic]).
11. Find the volume of the solid formed by rotating the area enclosed by [pic], [pic] and the [pic]-axis around the [pic]-axis.
Answers to Worksheet 4
1. [pic] 2. 5.9698 3. [pic] 4. [pic]
5. [pic] 6. [pic] 7. [pic] 8. [pic]
9. 8 10. 3.5864 11. 28.15
Worksheet 5
1. Let [pic] be the region in the first quadrant bounded by the graphs of [pic] and the line [pic].
a) Find the area of [pic] in terms of [pic].
b) Find the volume of the solid formed when [pic] is rotated 360 ° about the [pic]-
axis.
c) Find the volume in part b) as [pic].
2. Let [pic] be the shaded region in the first quadrant enclosed by the [pic]-axis and the graphs of [pic] and [pic] as shown in the figure below.
[pic]
a) Find the area of [pic].
b) Find the volume of the solid generated when [pic] is revolved about the
[pic]-axis.
c) Find the volume of the solid whose base is [pic] and whose cross sections
perpendicular to the [pic]-axis are squares.
3. Let [pic] be the region in the first quadrant under the graph of [pic] for [pic].
a) Find the area of [pic].
b) If the line [pic] divides the region [pic] into two regions of equal area, what
is the value of [pic]?
c) Find the volume of the solid whose base is the region [pic] and whose
cross-sections cut by planes perpendicular to the [pic]-axis are squares.
4. Let [pic] be the region enclosed by the graphs of [pic] and [pic].
a) Find the area of [pic].
b) The base of a solid is the region [pic]. Each cross-section of the solid
perpendicular to the [pic]-axis is an equilateral triangle. Find the volume of
the solid.
5. Find the volumes of the solids described:
a) The solid lies between planes perpendicular to the [pic]-axis at [pic] and
[pic]. The cross-sections perpendicular to the [pic]-axis are circular discs
with diameters running from the [pic]-axis to the parabola [pic].
b) The solid lies between planes perpendicular to the [pic]-axis at [pic] and
[pic]. The cross-sections perpendicular to the [pic]-axis between these
planes are squares whose diagonals run from the semicircle [pic]
to the semicircle [pic].
Answers to Worksheet 5
1. a) [pic] b) [pic] c) [pic]
2. a) 1.764 b) 30.46 c) 3.671
3. a) 2 b) [pic] c) 0.811 (approx.)
4. a) 1.168 b) 0.3967 (approx.)
5. a) [pic] b) [pic]
-----------------------
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
................
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