solids of revolution disk and ring method

1. Solids of revolution

2. The volume of a solid bounded by planes is easily defined in solid
geometry. However, for a solid bounded by curved surfaces or by a
combination of curved surfaces and planes we may obtain the volume by
evaluating a definite integral. Provided, the area of every plane section
parallel to some fixed plane can be expressed as a function of its distance
from the latter.
A solid of revolution is the figure formed when a plane region is
revolved about a fixed line. The fixed line is called the axis of revolution.
For short, we shall refer to the fixed line as axis or AOR.
The volume of a solid of revolution may be calculated using the
following methods: DISK, RING and SHELL METHOD
• Volume of a Solid
2

3. This method is used when the element (representative
strip) is perpendicular to and touching the axis. Meaning, the
axis is part of the boundary of the plane area. When the strip is
revolved about the axis of rotation a DISK is generated.
• A. DISK METHOD: V = πr2h
3
To find the volume of the entire solid:
𝑽 =
𝒂
𝒃
𝝅[𝒇 𝒙 ]𝟐 𝒅𝒙

4. • A. DISK METHOD: V = πr2h
4
𝑽 =
𝒂
𝒃
𝝅[𝒇 𝒙 ]𝟐 𝒅𝒙
𝑽 =
𝒄
𝒅
𝝅[𝒖 𝒚 ]𝟐 𝒅𝒚

5. 1. Sketch the bounded region and the line of revolution. (Make
sure an edge of the region is on the line of revolution.)
2. If the line of revolution is horizontal, the equations must be in
𝑦 = form. If vertical, the equations must be in 𝑥 = form.
3. Sketch a generic disk (a typical cross section).
4. Find the length of the radius and height of the generic disk.
5. Integrate with the following formula:
𝑉 = 𝜋
𝑎
𝑏
𝑟𝑎𝑑𝑖𝑢𝑠2 ∙ ℎ𝑒𝑖𝑔ℎ𝑡
Disk Method = No hole in the solid
• A. DISK METHOD: V = πr2h
5

6. Calculate the volume of the solid obtained by rotating the region
bounded by y = x2 and y=0 about the x-axis for 0 ≤ x ≤ 2.
• Example 1
6
Axis
of
Rotation
y = 𝑥2
x = 2
x = 0
y = 0

7. Calculate the volume of the solid obtained by rotating the region
bounded by y = x2, x=0, and y=4 about the y-axis.
• Example 2
7
Axis of Rotation
x = 2
x = 0
y = 4
y = 0

8. 11
Find the volume of the solid generated when the region
enclosed by 𝑦 = 𝑥 , 𝑦 = 6 − 𝑥 and 𝑦 = 0 is revolved about
the 𝑥-axis.
• Example 3

9. Ring or Washer method is used when the element (or
representative strip) is perpendicular to but not touching the
axis. Since the axis is not a part of the boundary of the plane
area, the strip when revolved about the axis generates a ring or
washer.
• B. RING or WASHER METHOD: V = π(𝑅2 − r2)h
9
𝑽 = 𝝅
𝒂
𝒃
𝒈 𝒙 𝟐 − 𝒇 𝒙 𝟐 𝒅𝒙 = 𝝅
𝒂
𝒃
𝒚𝒖
𝟐 − 𝒚𝑳
𝟐 𝒅𝒙

10. • B. RING or WASHER METHOD: V = π(𝑅2 − r2)h
10
𝑽 = 𝝅
𝒄
𝒅
𝒘 𝒚 𝟐 − 𝒗 𝒚 𝟐 𝒅𝒚 = 𝝅
𝒄
𝒅
𝒙𝑹
𝟐 − 𝒙𝑳
𝟐 𝒅𝒚

11. Area of a Washer:
The region between two concentric circles is called an annulus,
or more informally, a washer:
• B. RING or WASHER METHOD: V = π(𝑅2 − r2)h
11
𝑨𝒓𝒆𝒂 = 𝝅𝑹𝟐
𝒐𝒖𝒕𝒆𝒓 − 𝝅𝑹𝟐
𝒊𝒏𝒏𝒆𝒓
Rinner
Router 𝑨𝒓𝒆𝒂 = 𝝅 𝑹𝟐
𝒐𝒖𝒕𝒆𝒓 − 𝑹𝟐
𝒊𝒏𝒏𝒆𝒓

12. • Sketch the bounded region and the line of revolution.
• If the line of revolution is horizontal, the equations must be in
y= form. If vertical, the equations must be in x= form.
• Sketch a generic washer (a typical cross section).
• Find the length of the outer radius (furthest curve from the
rotation line), the length of the inner radius (closest curve to
the rotation line), and height of the generic washer.
• Integrate with the following formula:
• B. RING or WASHER METHOD: V = π(𝑅2 − r2)h
12
𝑽 = 𝝅
𝒂
𝒃
𝑹𝟐
𝒐𝒖𝒕𝒆𝒓 − 𝑹𝟐
𝒊𝒏𝒏𝒆𝒓 ∙ 𝒉𝒆𝒊𝒈𝒉𝒕
Washer Method = Hole in the
solid

13. Calculate the volume V of the solid obtained by rotating the region
bounded by 𝑦 = 𝑥2 and 𝑦 = 0 about the line 𝑦 = −2 for 0 ≤ 𝑥 ≤ 2.
• EXAMPLE 4

14. • EXAMPLE 5
14

15. Determine the volume of the solid generated by the region between 𝑦 =
𝑥2 and 𝑦 = 4𝑥, revolved about the 𝑦-axis.
• EXAMPLE 6
15
y = 4x
y = 𝑥2