Section 7.2 – Volume: The Disk Method

White Board Challenge

Find the volume of the following cylinder:

12 ft

6 ft

2 33 12 108 339.292V ft

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White Board ChallengeCalculate the volume V of the solid obtained by rotating the region between y = 5 and the x-axis about the x-axis for 1≤x≤7.

150V

471.239V

2V radius height

5

6

25 6V

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alculator

Volumes of Solids of Revolution with Riemann Sums

The Riemann Sum is set up by considering this cross sections of the solid (circles) each with thickness dx:

kxVolume

2kradius x

1

n

kmax 0

limkx

2b

aradius dx a b

Radius

2b

aradius dx

Volumes of Solids of Revolution: Disk Method

• Sketch the bounded region and the line of revolution. (Make sure an edge of the region is on 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 disk (a typical cross section).• Find the length of the radius and height of the

generic disk.• Integrate with the following formula:

2b

aV radius height

Disk Method = No hole in the

solid.

Example 1Calculate 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.

Sketch a GraphFind the Boundaries/Intersections

0,2x

Make Generic Disk(s)

Height = dx

Radius = x2

Integrate the Volume of Each Generic Disk

2 22

0x dx325

Line

of R

otat

ion

Example 2Calculate the volume of the solid obtained by rotating the region bounded by y = x2 and y=4 about the line y = 4.

Sketch a GraphFind the Boundaries/Intersections

2,2x

Make Generic Disk(s)

Height = dx

Radius = 4 - x2

Integrate the Volume of Each Generic Disk

2 22

24 x dx

51215

Line

of R

otat

ion

NOTE: Because of the

square, the order of

subtraction does not matter.

2 4x

Example 3Calculate the volume of the solid obtained by rotating the region bounded by y = x2, x=0, and y=4 about the y-axis.

Sketch a Graph

Find the Boundaries/Intersections

Make Generic Disk(s)

Height = dy

Radius = √y

Integrate the Volume of Each Generic Disk

24

0y dy 8

Line of Rotation

Since the Line of Revolution is Vertical, Solve for x

2y xx yx

We only need 0≤x≤2

0y

20y 4y Remember: 0≤x≤2

White Board ChallengeFind the volume of the following three-dimensional

shape:

12 ft

6 ft 2 ft

2 2

2 2

3

3 12 1 12

3 1 12

96 301.593

V

ft

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alculator

6

White Board ChallengeCalculate the volume V of the solid obtained by rotating the region between y = 5 and the y = 2 about the x-axis for 1≤x≤7.

126V

395.841V

2 2outer innerV r h r h

5

625 V

2 2outer innerV r r h

222

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Area of a WasherThe region between two concentric circles is called an annulus, or more informally, a washer:

Rinner

Router

2 2outer innerArea R R

2 2outer innerArea R R

Volumes of Solids of Revolution: Washer Method

• 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:2 2b

outer inneraV r r height

Washer Method = Hole in the

solid.

Always a difference of squares.

Example 1Calculate 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.

Sketch a GraphFind the Boundaries/Intersections

0,2x

Make Generic Washer(s)Height = dx

Router = x2 - -2 = x2 + 2 Integrate the Volume of Each

Generic Washer

2 22 2

02 2x dx

25615

Line

of R

otat

ion

Rinner = 0 - -2 = 2

Example 2Calculate the volume V of the solid obtained by rotating the region bounded by y = ex and y=√(x +2) about the line y = 2.

Sketch a GraphFind the Boundaries/Intersections

2xe x

Make Generic Washer(s)

Height = dx

Router = 2 - ex

Integrate the Volume of Each Generic Washer

20.448 2

1.9812 2 2xe x dx

8.536

Line

of R

otat

ion

Rinner = 2- √(x +2)

1.981, 0.448x

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“Warm-up”: 1985 Section I

CAN DO NOW:

Volume of a Right SolidA right solid is a geometric solid whose sides are perpendicular to the base. The volume of a right solid is the area of the base times the height.

HSolid

ABase

Base SolidVolume A H

Volumes of Solids: Slicing Method

• Sketch the bounded region. • If the cross section is perpendicular to the x-axis,

the equations must be in y= form. If the y-axis, the equations must be in x= form.

• Sketch a generic slice (a typical cross section).• Find the area of the base and the height of the

generic slice.• Integrate with the following formula:

b

BaseaV A height

Must Answer #1: What does the length

across the bounded region represent in your generic slice?

Must Answer #2: How does the length across the bounded region help find the area of the base of the generic slice?

Example 1Find the volume of the solid created on a region who base is bounded by y = √x and the x-axis for 0≤x≤9. Let each cross section be perpendicular to the x-axis and be a square.

Sketch a Graph Find the Boundaries/Intersections0,9x

Height = dxABase

= ASquare

= side2

Integrate the Volume of Each Generic Slice

29

0x dx812

Make Generic Slice(s)

= (√x)2

Side

Len

gth

Example 2Find the volume of the solid created on a region who base is bounded by x2 + y2 = 1. Let each cross section be perpendicular to the x-axis and be a squares with diagonals in the xy-plane.

Sketch a Graph

Find the Boundaries/Intersections

1,1x

Height = dx

ABase = ASquare

= side2

Integrate the Volume of Each Generic Slice

221 2 1

21

x dx

83

Make Generic Slice(s)

Since the Cross Sections are Per. to the x-axis, solve for y

2 2 1x y 21y x

d

2d

2 2 21 1

2x x

22

2 12

x

If diameter is known, a side

length is…

Dia

gona

l

White Board ChallengeA solid has base given by the triangle with vertices (-4,0), (0,8), and (4,0). Cross sections perpendicular to the y-axis are semi-circles with diameter in the plane.

What is the volume of the solid?

Calculator

12 4x y

12 4x y

Radius = -½y+4ABase

= ½πr2

Height = dy

643

8 21 1

2 204y dy

Diameter