The experimental verification of a theory
concerning any natural phenomenon
generally rests on the result of an
integration…
J. W. Mellor…
Area of a Region Between Two Curves
AP Calculus
§ Area of a Region Between two Curves
If f and g are continuous on [a, b] and g (x)  f (x) for
all x in [a, b], then the area of the region bounded by the
graphs of f and g and the vertical lines x = a and x = b
is
A

f ( x)  g ( x) dx
a
b
(Upper Function – Lower Function)
Larson – Hostetler – Edwards: Chapter 7.1
Area of a Region Between Two Curves
AP Calculus
§ Area of a Region Between two Curves
f (x)

b
a

f ( x) dx
b
a
g ( x) dx
g (x)
Larson – Hostetler – Edwards: Chapter 7.1
Area of a Region Between Two Curves
AP Calculus
§ Area of a Region Between two Curves
f (x)
 f a( xf) (xg) (dxx) dx
b
b
a

b
a
g (x)
g ( x) dx
Larson – Hostetler – Edwards: Chapter 7.1
Area of a Region Between Two Curves
AP Calculus
§ Example.
Find the area between the two given curves
(a). y  x
yx
2
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
Area of a Region Between Two Curves
AP Calculus
§ Example.
Find the area between the two given curves
(b). y  x  1
 x2
y  xe
x  0, x  2
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
AP Calculus
§ Example.
Area of a Region Between Two Curves
Find the area between the two given curves
(c). y  4 x  16
y  2 x  10
2
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
Area of a Region Between Two Curves
AP Calculus
§ Example.
Find the area between the two given curves
(d ). y  4 x  16 y  2 x  10
2
x  2, x  5
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
AP Calculus
§ Example.
Area of a Region Between Two Curves
Find the area between the two given curves
(e). y  cos x y  sin x
x  0, x   / 2
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
AP Calculus
§ Example.
Area of a Region Between Two Curves
Find the area between the two given curves
1 2
( f ). x  y  3 x  y  1
2
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
AP Calculus
§ Example.
Area of a Region Between Two Curves
Find the area between the two given curves
( g ). x   y  10 x  ( y  2)
2
2
Paul's
Online Math
Notes
Larson –From:
Hostetler
– Edwards:
Chapter
7.1
AP Calculus
Area of a Region Between Two Curves
Practice. Textbook 7.1 pp. 452-455, prob. 1, 3, 5, 13, 17,
21, 23, 27, 33, 37, 39, 43, 45, 47
Suggested Extra Practice : pp 452-455, prob. 2, 4, 6,
14, 18, 22, 24, 28, 34, 38, 40,
44, 46, 48
Larson – Hostetler – Edwards: Chapter 7.1
The moving power of mathematical
invention is not reasoning but
imagination …
Augustus de Morgan…
AP Calculus
Volume: The Disk Method
§ Solids of Revolution
This is a solid obtained by rotating a region in the plane
about an axis (called axis of revolution).
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
§ Solids of Revolution
This is a solid obtained by rotating a region in the plane
about an axis (called axis of revolution).
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Solids of Revolution
w
r
w
r
The simplest solid of revolution is a
right circular cylinder or disk, which is
formed by revolving a rectangle about
an adjacent axis
The corresponding volume is:
Volume disk = area disk  width disk
V = r2 w
r = radius of the disk
w = width
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
§ Solids of Revolution
To find the volume of any
solid of revolution, consider
this as composed of n-disks
of width w = Dx and radius
r = r ( xi )
An approximation for the
volume of the solid will be:
n
V    [r ( xi )] Dx
2
i 1
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
§ Solids of Revolution
Increasing the number n of
disks ( i. e. Dx  0 ), the
approximation to the volume
becomes better.
The volume of the solid will
be given by the limit process:
n
V  lim   [r ( xi )] Dx
2
D0
b
i 1
   [r ( x)] dx
2
a
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
§ The Disk Method
If r(x) is continuous and r(x) > 0 on [a, b], then the
volume obtained by rotating the region under the graph,
can be calculated using one of the following formulas.
Horizontal Axis of Revolution:
b
V    [r ( x)] dx
2
a
Vertical Axis of Revolution:
b
V    [r ( y )] dy
2
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Show that the volume of a sphere of radius r is
4 3
V  r
3
y
r
0
x
r
y
y r x
2
b
2
V    [ y ( x)] dx
2
a
x
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Show that the volume of a right cone of radius r and height h is
given by the formula :
y
r
y
0
h
x
1 2
V  r h
3
r
y x
h
b
V    [ y ( x)] dx
2
a
x
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
§ Example. Find the volume of the solid obtained by rotating about the x-axis
the region between x = 0, x = 1, and under the curve
y x
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating about the x-axis
the region between the lines x = 1, x = 3, and under the curve
1
y
x
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating the region
bounded by y = x 3, y = 8, and x = 0 about the y-axis
yx
1
3
x y 3 y
3
b
V    [ x( y )] dy
2
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating about the y-axis
the region bounded by x = 4, x = 0 and
y x
x y
b
2
V    [ x( y )] dy
2
a
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
Practice. Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7,
9, 11, 13
Suggested Extra Practice : pp 463-466, prob. 2, 4,
6, 8, 10, 12, 14
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
§ The Washer Method
Method useful to find the volume of a solid of revolution
with a hole, by changing the disk with a washer. The
washer is formed by revolving a rectangle between to
curves around the x-axis or any axis of the form x = a
w
w
Volume
of
R
r
Washer
R
r
V = (R2 – r2)w
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ The Washer Method
If R(x) and r(x) are continuous functions on [a, b],
R(x) > 0, r(x) > 0, and R(x) > r(x), then the volume
obtained by rotating the region between R(x) and r(x)
can be calculated using one of the following formulas:
Horizontal Axis
of Revolution:
Vertical Axis of
Revolution:
b


V    [ R( x)]  [r ( x)] dx
a
b

2
2

V    [ R( y )]  [r ( y )] dy
a
2
2
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating the region
between the lines f(x) = x2 + 3, and g(x) = x2 + 1, around the
x- axis
r ( x)  x 2  1
R( x)   x 2  3
b


V    [ R( x)]  [r ( x)] dx
a
2
2
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating the region
between the lines f(x) and g(x), around the x- axis
f ( x)  2 
b

1 x2
g ( x)  2

V    [ R( x)]2  [r ( x)]2 dx
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating the region
between the lines f(x) and g(x), around the axis y = 1
f ( x)  x 2  2
b

g ( x)  4  x 2

V    [ R( x)]2  [r ( x)]2 dx
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating the region
between the lines f(x), g(x) and x = 1, around the axis y = 2
f ( x)  x  4
b

g ( x)   x / 3

V    [ R( x)]2  [r ( x)]2 dx
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid formed by revolving the region
bounded by the graph of y = x2 + 1, y = 0 , x = 0 and x = 1,
about the y - axis
b


V    [ R( y )]2  [r ( y )]2 dy
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid formed by revolving the region
bounded by the graph of y = 3/x, and y = 4  x, about x =  1
b


V    [ R( y )]2  [r ( y )]2 dy
a
Larson – Hostetler – Edwards: Chapter 7.2
AP Calculus
Volume: The Disk Method
Practice. Textbook 7.2 pp. 463-466, prob. 15, 17, 19, 21,
23, 27, 29
Suggested Extra Practice : pp 463-466, prob. 16,
18, 20, 22, 24, 28, 30
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Solids with Known Cross Sections
With the disk method it is possible to find the volume of a
solid with a circular cross section:
R(x)]2 = A(x)  V  
b
 [ R( x)] dx  
a
2
b
a
A( x) dx
This method can be generalized to solids with any shape,
as long as a formula for the area of the arbitrary cross
section is known
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Solids with Known Cross Sections
Area of Cross Section = A(x)
n
V   Ai ( x) Dx
i 1
V  lim
n
Dx 0
 A ( x)Dx
i 1
i
b
V   A( x) dx
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Volumes of Solids with Known Cross Sections
1. For cross sections of area A(x) taken perpendicular to
the x-axis:
b
V   A( x) dx
a
2. For cross section of area A(y) taken perpendicular
to the y-axis
b
V   A( y ) dy
a
Larson – Hostetler – Edwards: Chapter 7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid with semicircular cross section and
whose base is bounded by the graphs of y = 0, x = 9 and
y x
x
r
2
r
y x
1  x

A( x)   
2  2 
2
b
V   A( x) dx
a
From: DEMOS with
POSITIVE
IMPACT,
NSF DUEChapter
9952306 7.2
Larson
– Hostetler
– Edwards:
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid with square cross section and whose
base is bounded by the graphs of y  x / 3, y   x / 3 , x  1
 2x 
A( x)  

 3
2x
3
2
b
2x
3
V   A( x) dx
a
From: DEMOS with POSITIVE IMPACT
David R. Hill, Temple University
F. Roberts, Georgia
College & State
University
Larson- Lila
– Hostetler
– Edwards:
Chapter
7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid with equilateral triangle cross section
and base bounded by the graphs of y = sin x, x = 0, x = 
1
A( x)  sin
2
 3

x
sin x 
 2

b
V   A( x) dx
a
sin x
From: DEMOS with POSITIVE IMPACT
David R. Hill, Temple University
F. Roberts, Georgia
College & State
University
Larson- Lila
– Hostetler
– Edwards:
Chapter
7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid with square cross section and base
bounded by the graphs of x = 0, x = 1, y = 0 and y  1  x 2

A( x)  1  x
1 x2
2

2
b
1 x
2
V   A( x) dx
a
From: DEMOS with POSITIVE IMPACT
David R. Hill, Temple University
F. Roberts, Georgia
College & State
University
Larson- Lila
– Hostetler
– Edwards:
Chapter
7.2
Volume: The Disk Method
AP Calculus
§ Example. Find the volume of the solid with equilateral triangle cross section
and base bounded by the circle of radius 2, centered at (0, 0)
y
y  4  x2
3 4  x2
x
y   4  x2

A( x)  3 4  x
2

2 4  x2
b
V   A( x) dx
a
From: DEMOS with POSITIVE IMPACT
David R. Hill, Temple University
F. Roberts, Georgia
College & State
University
Larson- Lila
– Hostetler
– Edwards:
Chapter
7.2
AP Calculus
Volume: The Disk Method
Practice. Textbook 7.2 pp. 463-466, prob. 61, 62, 63, 64
Suggested Extra Practice : pp 463-466, prob. 61,
62, 63, 64
Larson – Hostetler – Edwards: Chapter 7.2
The advancement and perfection of
mathematics are intimately connected
with the prosperity of the State …
Napoleon…
Volume: The Shell Method
AP Calculus
§ The Shell Method
This is an alternative method for finding the volume of a
solid of revolution
Compared with the disk method, the representative
rectangular section is parallel to the axis of revolution
w
h
w
h
w
r
Disk Method
r
p
p
Shell Method
Larson – Hostetler – Edwards: Chapter 7.3
AP Calculus
Volume: The Shell Method
§ The Shell Method
Shell Method
Disk Method
From: DEMOS with POSITIVE IMPACT
Larson – Hostetler – Edwards: Chapter 7.3
Volume: The Shell Method
AP Calculus
§ The Shell Method
y
h
y
h
p + w/2
w
w
p
p
p  w/2
x
x
2
2
Volume of   p  w  h    p  w  h  2  p h w




Shell:
2
2


If w = Dy, p = p(y), h = h(y)

DV = 2[ p(y) h(y)]Dy
Larson – Hostetler – Edwards: Chapter 7.3
Volume: The Shell Method
AP Calculus
§ The Shell Method
The volume of the solid obtained by revolving the area
under function h, can be calculated in the next form:
Vertical
axis of
revolution
b
V  2  p( x)h( x)dx
a
p(x) = distance to axis
Horizontal V  2 d p( y )h( y )dy
c
axis of
p(y) = distance to axis
revolution

Larson – Hostetler – Edwards: Chapter 7.3
Volume: The Shell Method
AP Calculus
§ Example. Find the volume of the solid obtained by rotating the area between
the graph of f(x) = 1 – 2x – 3x2 – 2x3 and the x-axis over [0, 1]
about: (a) the y-axis, (b) x = 1
(b)
(a)
b
V  2  p( x)h( x)dx
a
Rogawski,
4027.3
Larson – Hostetler – Edwards:
Chapter
Volume: The Shell Method
AP Calculus
§ Example. Calculate the volume of the solid obtained by rotating the area
enclosed by the graphs of f(x) = 9 – x2 and g(x) = 9 – 3x
about: (a) x-axis, (b) y = 10
(a)
(b)
Rogawski,
Larson – Hostetler – Edwards:
Chapter403
7.3
Volume: The Shell Method
AP Calculus
§ Example. Calculate the volume of the solid obtained by rotating the region
between the graph f (y) = 12(y2 – y3) and the y-axis, over [0, 1],
about: (a) the x-axis, (b) the line y = 1
(a)
(b)
Finney, 392
Larson – Hostetler – Edwards: Chapter
7.3
Volume: The Shell Method
AP Calculus
§ Example. Calculate the volume of the solid obtained by rotating the region
between the graphs of f (x) = y 4/4 – y 2/2 and x = y 2/2 about:
(a) x-axis, (b) y = 2
(a)
(b)
Stewart 395
Larson – Hostetler – Edwards: Chapter
7.3
Volume: The Shell Method
AP Calculus
§ Example. Calculate the volume of the solid obtained by rotating the region
between the graph f (x) = x sin x and the x-axis, over [0, ]
about: (a) the y-axis, x = 2
(a)
(b)
Stewart 396
Larson – Hostetler – Edwards: Chapter
7.3
Volume: The Shell Method
AP Calculus
§ Example. Calculate the volume of the solid obtained by rotating the region
between the graph f (x) = sin x and:
(a) the x-axis, over [0, 4] about the line x = 5
(b) the line y = 4, over [0, 4], about the y-axis
(a)
(b)
Larson – Hostetler – Edwards: Chapter 7.3
Volume: The Shell Method
AP Calculus
§ Example. Calculate the volume of the solid obtained by rotating the circle
(x – 2)2 + y2 = 1 about the y-axis
Larson – Hostetler – Edwards: Chapter 7.3
AP Calculus
Volume: The Shell Method
Practice. Textbook 7.2 pp. 463-466, prob. 1, 3, 5, 7, 9, 11,
14, 15, 17, 19
Suggested Extra Practice : pp 463-466, prob. 2, 4, 6,
8, 10, 12, 14, 16, 18, 20
Larson – Hostetler – Edwards: Chapter 7.3
Volume: The Shell Method
AP Calculus
§ The Shell Method

d
The volume of the solid obtainedVbyrevolving
2 p( the
y )harea
( y )dy
under function h, can be calculated in the cnext form:
p(y) = distance to axis
Vertical
axis of
revolution
Horizontal
axis of
revolution
b
V  2  p( x)h( x)dx
a
p(x) = distance to axis
Larson – Hostetler – Edwards: Chapter 7.3
Volume: The Shell Method
AP Calculus
§ The Shell Method

b
The volume of the solid obtainedVbyrevolving
2 p( xthe
)h(area
x)dx
under function h, can be calculated in theanext form:
p(x) = distance to axis
Vertical V  2
p( y )h( y )dy
c
axis of
p(y) = distance to axis
revolution

d
Horizontal
axis of
revolution
Larson – Hostetler – Edwards: Chapter 7.3