Chapter 6. Additional Applications of the
Integration
Le Cong Nhan
Faculty of Applied Sciences
HCMC University of Technology and Education
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Contents
1
Areas Between Two Curves
Using Vertical Strips
Using Horizontal Strips
2
Volume
Method of Cross Sections
Volume of a Solid of Revolution
Methods of Disks and Washers
Method of Cylindrical Shells
3
Polar Forms and Area
4
Arc Length and Surface Area
5
Physical Applications: Work, Liquid Force, and Centroids
Work
Liquid Force
Centroids
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6.1 Areas Between Two Curves
Areas Between Two Curves
Find the area of the region R bounded by the curves y = f (x) and y = g (x) and
the lines x = a, x = b, where f and g are continuous and f (x) ≥ g (x) for all
x ∈ [a, b].
The area A of the region R is given by
Z
b
[f (x) − g (x)] dx
A=
(1)
a
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Area Using Vertical Strips
Find the area of the region R that lies between two curves y = f (x) and y = g (x)
from x = a to x = b, where f and g are continuous. Then its area is
Z
b
|f (x) − g (x)| dx
A=
(2)
a
Figure: The region R = R1 ∪ R2 ∪ R3
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Area Using Horizontal Strips
If a region R is bounded by curves with equations x = f (y ), x = g (y ), y = c, and
y = d, where f and g are continuous, then its area is
Z
d
|f (y ) − g (y )| dy
A=
(3)
c
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Example 1
(a) Find the area of the region bounded above by y = e x , bounded below by
y = x, and bounded on the sides by x = 0 and x = 1.
(b) Find the area of the region between the curves y = x 3 and y = x 2 − x on the
interval [0, 1].
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Example 2
(a) Find the area of the region enclosed by the parabolas y = x 2 and y = 2x − x 2 .
(b) Find the area of the region bounded by the curve y = e 2x − 3e x + 2 and the
x-axis.
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Example 3
(a) Find the area of the region bounded by y = x 2 , y = x, x = −1 and x = 1.
(b) Find the area of the region bounded by the line y = 3x and the curve
y = x 3 + 2x 2 .
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Example 4
(a) Find the area of the region bounded by the curve y = sin x and the x-axis
between x = −π/2 to x = π/2.
(b) Find the area of the region bounded by the curves y = sin x, y = cos x and
between x = 0 to x = π/2.
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Example 5
(a) Find the area enclosed by the line y = x − 1 and the parabola y 2 = 2x + 6.
(b) Find the area enclosed by the line x = 2y − 3 and the parabola x = 4y − y 2 .
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6.2 Volume
Volume of a cylinder
If the area of the base is A and the height of the cylinder is h, then the volume V
of the cylinder is defined as
V = Ah
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How to find the volume of a solid S that isn’t a cylinder?
1. Cutting S into pieces and approximate each piece by a cylinder
2. Estimate the volume of S by adding the volumes of the cylinders
3. The exact volume of S is obtained by a limiting process in which the number
of pieces becomes large
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Method of Cross Sections
A solid S with cross-sectional area A(x) in the plane Px perpendicular to the x-axis
at each point x on the interval [a, b] has volume
Z
V =
b
A(x)dx
(4)
a
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Example 6
Show that
4 3
πr .
3
(b) the volume of a pyramid whose base is a square with side L and whose height
1
is h is V = L2 h.
3
(a) the volume of a sphere of radius is V =
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The Volume of a Solid of Revolution
Definition 7 (Solid of revolution)
A solid of revolution is a solid obtained by revolving a region D in the xy -plane
about a line L (called the axis of revolution) that lies outside or on the boundary
of D.
Figure: Solids of revolution
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The Volume of a Solid of Revolution
Methods of Disks and Washers
The volume of a solid of revolution
Z
V =
b
A(x)dx
(5)
a
If the cross-section is a disk, we find the radius of the disk and use
2
A = π (radius)
If the cross-section is a washer (an annular ring), we find the inner radius rin
and outer radius rout and compute the area of the washer by
h
i
2
2
2
2
A = π (rout ) − π (rin ) = π (rout ) − (rin )
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The Disk Method
Let D be the region under the curve y = f (x), above the x-axis from a to b. Then
the volume V of the solid S obtained by rotating D about the x-axis is
Z
V =
b
2
π [f (x)] dx
(6)
a
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Example 8
(a) Find the volume of the
√ solid obtained by rotating about the x-axis the region
under the curve y = x from 0 to 1.
(b) Find the volume of the solid S formed by revolving the region D under the
curve y = x 2 + 1 on the interval [0, 2] about the x-axis.
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The Washer Method
Let D be the region bounded by y = f (x), y = g (x) with f (x) ≥ g (x) ≥ 0, from a
to b. Then the volume V of the solid S obtained by rotating D about the x-axis
is
Z b 2
2
V =
π [f (x)] − [g (x)] dx
(7)
a
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Example 9
Let D be the solid region bounded by the parabola y = x 2 and y = x. Find the
volume of the solid generated when D is revolved about
a. the x-axis
b. the y -axis
c. the line y = 2
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The Volume of a Solid of Revolution
Methods of Cylindrical Shell
The volume of a cylindrical shell
V = 2πrh∆r
= 2π × ravg × height × thickness
where r =
shell.
r1 + r2
is the average radius of the
2
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Method of Cylindrical Shells
Let D be the region bounded by y = f (x) (where f (x) ≥ 0), y = 0, x = a, and
x = b, where b > a ≥ 0. Then the volume V of the solid S obtained by rotating
D about the y -axis is
Z
V =
b
2πxf (x)dx
(8)
a
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Sketch the Idea of the Shell Method
1. We divide the interval [a, b] into n subintervals [xi−1 , xi ] of equal width ∆x
and let x¯i be the midpoint of the ith subinterval. If the rectangle with base
[xi−1 , xi ] and height f (x¯i ) is rotated about the y -axis, then the result is a
cylindrical shell with average radius x¯i , height f (x¯i ), and thickness ∆x, so its
volume is
Vi = (2π x¯i ) [f (x¯i )] ∆x
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2. An approximation to the volume V of S is given by
V ≈
n
X
i=1
Vi =
n
X
2π x¯i f (x¯i )∆x
i=1
3. Let n → ∞, we get the volume V of S
V = lim
n→∞
n
X
i=1
Z
2π x¯i f (x¯i )∆x =
b
2πxf (x)dx
a
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Example 10
(a) Find the volume of the solid formed by revolving the region bounded by the
x-axis and the graph of y = x 3 + x 2 + 1, x = 1, and x = 3 about the y -axis.
(b) Find the volume of the solid obtained by rotating about the y -axis the region
bounded by y = 2x 2 − x 3 and y = 0.
(c) 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.
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6.3 Polar Forms and Area
Polar coordinate system
P 7→ P(r , θ): polar coordinates
r = OP is the radial coordinate of P
θ is the polar angle of P
Convention:
Figure: Polar coordinate system
θ > 0: the counterclockwise direction
from the polar axis
θ < 0: the clockwise direction
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Changing Coordinates
(1) If P(r , θ) then P(x, y ) with
x = r cos θ,
y = r sin θ
(2) If P(x, y ) then P(r , θ) with
r=
p
x 2 + y 2,
tan θ =
y
, x 6= 0
x
We extend the meaning of polar coordinates
(r , θ) to the case r < 0 as follows:
if r > 0 the point (r , θ) lies in the same
quadrant as θ;
Figure: Extended polar coordinate
system
if r < 0 it lies in the quadrant on the
opposite of the pole.
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6.3 Polar Forms and Area
Polar Curves
In this section, we will study:
Polar Forms
Polar curves
Intersection of polar curves
Polar area
The graph of a polar equation
r = f (θ)
or F (r , θ) = 0.
Example 11
Sketch the polar curve r = 2 and θ =
π
6
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Example 12
(a) Sketch the curve with polar equation r = 2 cos θ.
(b) Find a Cartesian equation for this curve.
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Symmetry of Polar curves
(a) If a polar equation is unchanged when θ is replaced by −θ, the curve is symmetric about the polar axis.
(b) If the equation is unchanged when r is replaced by −r , or when θ is replaced
by θ + π, the curve is symmetric about the pole. (This means that the curve
remains unchanged if we rotate it through 1800 about the origin.)
(c) If the equation is unchanged when θ is replaced by π−θ, the curve is symmetric
about the vertical line θ = π/2.
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Example 13
Sketch the curve
a. r = 1 + cos θ
b. r = cos 2θ
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Intersection of Polar-Form Curves
Step 1: Find all simultaneous solutions of given system of equations.
Step 2: Determine whether the pole r = 0 lies on the two graphs.
Step 3: Graph the curve to look for other points of intersection.
Example 14
Find the points of intersection of the curves r =
2π
3
− cos θ and θ =
.
2
3
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6.3 Polar Forms and Area
Polar Area
The area of a circular sector of radius r is
given by
A=
1 2
r θ
2
where r is the radius and θ is the radian
measure of the central angle.
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Area in Polar Coordinates (Case 1)
Let R be the region bounded by the polar curve r = f (θ) and by the rays θ = a
and θ = b, where f is a positive continuous function and where 0 < b − a ≤ 2π.
A=
Figure: Polar area
1
2
Z
b
2
[f (θ)] dθ
(9)
a
Figure: Approximating sector
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Area in Polar Coordinates (Case 2)
Let R be the region bounded by curves with the polar equations r = f (θ), r = g (θ)
and by the rays θ = a and θ = b, where f (θ) ≥ g (θ) ≥ 0 and 0 < b − a ≤ 2π.
The area A of R is
Z
1 b
2
2
(10)
[f (θ)] − [g (θ)] dθ
A=
2 a
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Example 15
(a) Find the area of the top half (0 ≤ θ ≤ π) of the cardioid r = 1 + cos θ.
(b) Find the area enclosed by one loop of the four-leaved rose r = cos 2θ.
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Example 16
(a) Find the area of the region that lies inside the circle r = 3 sin θ and outside
the cardioid r = 1 + sin θ.
(b) Find the area of the region common to the circles r = a cos θ and r = a sin θ.
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Example 17
(a) Find the area between the circle r = 5 cos θ and the limacon r = 2 + cos θ.
Round your answer to the nearest hundredth of a square unit.
(b) Find the area of the region that lies inside the circle r = 3 sin θ and outside
the cardioid r = 1 + sin θ.
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6.4 Arc Length and Surface Area
The Arc Length of a Curve
The Arc Length Formula (Case 1)
If f 0 is continuous on [a, b], then the length of the curve y = f (x), a ≤ x ≤ b, is
Z
L=
b
q
2
1 + [f 0 (x)] dx
(11)
a
The length L of the curve C
L = lim
n→∞
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n
X
|Pi−1 Pi |
j=1
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The Arc Length Formula (Case 2)
If a curve has the equation x = g (y ), c ≤ y ≤ d and g 0 (y ) is continuous on [c, d],
then
Z dq
2
L=
1 + [g 0 (y )] dy
(12)
c
Example 18
(a) Find the arc length (rounded to two decimal places) of the curve y = x 3/2 on
the interval [0, 4].
(b) Find the length of the curve defined by y = sin x on [0, 2π].
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Example 19
(a) Find the arc length of the curve x = 31 y 3 + 41 y −1 from y = 1 to y = 3.
(b) Find the length of the arc of the parabola y 2 = x from (0, 0) to (1, 1).
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6.4 Arc Length and Surface Area
The Area of a Surface of Revolution
The area of the band (or frustum of a cone) with
slant height l and upper and lower radii r1 and r2
is
S = πr2 (l1 + l) − πr1 l1
= π [(r2 − r1 ) l1 + r2 l]
From similar triangles we have
Figure: The area of the band
r1 + r2
S = 2πrl, r =
2
l1 + l
l1
=
⇒ (r2 − r1 ) = r1 l
r1
r2
The area of the band
S = π (r1 l + r2 l) = 2πrl
where r =
(13)
r1 + r2
is the average radius of the band.
2
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Surface Area (Case 1)
Suppose f is positive and has a continuous derivative on [a, b], we define the
surface area of the surface obtained by rotating the curve y = f (x), a ≤ x ≤ b,
about the x-axis as
Z b
S=
2πyds
a
Z
=
b
q
2
2πf (x) 1 + [f 0 (x)] dx
(14)
a
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Surface Area
Suppose f is positive and has a continuous derivative on [a, b], we define the
surface area of the surface obtained by rotating the curve y = f (x), a ≤ x ≤ b,
about the y -axis as
Z
b
S=
2πxds
a
Z
=
b
q
2πx
2
1 + [f 0 (x)] dx
(15)
a
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Surface Area (Case 2)
If the curve is described as x = g (y ), c ≤ y ≤ d, then the formula for surface area
obtained by rotating the curve x = g (y ), c ≤ y ≤ d, about the y -axis becomes
Z
S=
d
q
2
2πg (y ) 1 + [g 0 (y )] dy
(16)
c
Example 20
√
The curve y = 4 − x 2 , −2 ≤ x ≤ 2, is an arc of the circle x 2 + y 2 = 4. Find the
area of the surface obtained by rotating this arc about the x-axis.
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Example 21
The arc of the parabola y = x 2 from (1, 1) to (2, 4) is rotated about the y -axis.
Find the area of the resulting surface.
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Example 22
Find the area of the surface generated by rotating the curve y = e x , 0 ≤ x ≤ 1,
about the x-axis.
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Polar Arc Length and Surface Area
Polar Arc Length
Arc Length in Polar Coordinates
The length of a polar curve r = f (θ) for a ≤ θ ≤ b is given by the integral
s
2
Z b
dr
L=
dθ
r2 +
dθ
a
(17)
It is noticed that
q
2
2
∆s = (r ∆θ) + (∆r )
s
2
∆r
2
= r +
∆θ
∆θ
Figure: Polar arc length
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Example 23
(a) Find the length of the circle r = 2 sin θ
(b) Find the length of the polar curve r = cos θ
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Example 24
Find the length of the polar curve
a. r = e 3θ ,
0≤θ≤
π
2
b. r = cos2
θ
2
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Polar Arc Length and Surface Area
Polar Surface Area
Surface area in Polar Coordinates
If a polar curve r = f (θ) for a ≤ θ ≤ b is revolved about the x-axis, it generates
a surface of area
Z b
S=
2πyds
a
s
2
Z b
dr
2
=2π
(r sin θ) r +
dθ
(18)
dθ
a
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Example 25
Find the area of the surface generated by revolving about the x-axis the top half
of the cardioid r = 1 + cos θ.
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Example 26
Find the surface area generated when the given polar curve is revolved about the
x-axis
θ
a. r = 1 − cos θ, 0 ≤ θ ≤ π
b. r = cos2 , 0 ≤ θ ≤ π
2
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6.5 Physical Applications: Works, Liquid Force, and
Centroids
If a body moves a distance d in the direction of an applied constant force F , the
work W done is
W = Fd
Mass
Distance
Force
Work
kg
m
newton(N)
joule
g
cm
dyne(dyn)
erg
slug
ft
pound
ft-lb
Table: Common units of work and force
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Work done by a variable force
The work done by the variable force F (x) in moving an object along the x-axis
from x = a to x = b is given by
Z
b
F (x)dx
W =
(19)
a
Example 27
An object located x ft from a fixed starting position is moved along a straight
road by a force of F (x) = 3x 2 + 5 lb. What work is done by the force to move
the object
(a) through the first 4 ft?
(b) from 1 ft to 4 ft?
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Hooke’s law
When a spring is pulled x units past its equilibrium (rest) position, there is a
restoring force
F (x) = kx
that pulls the spring back toward equilibrium. The constant k is so-called the
spring constant.
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Example 28 (Modeling Work using Hooke’s Law)
The natural length of a certain spring is 10 cm. If it requires 2 ergs of work to
stretch the spring to a total length of 18 cm, how much work will be performed in
stretching the spring to a total length of 20 cm?
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Example 29 (The Work Performed in Pumping Water Out of a Tank)
A tank in the shape of a right circular cone of height 12 ft and radius 3 ft is
inserted into the ground with its vertex pointing down and its top at ground level,
as shown in the figure below. If the tank is filled with water (weight density
δ = ρg = 62.4 lb/ft3 ) to a depth of 6 ft, how much work is performed in pumping
all the water in the tank to ground level? What changes if the water is pumped to
a height of 3 ft above ground level?
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Example 30 (Exercise 37, p. 482)
A tank in the shape of an inverted right circular cone of height 6 ft and top radius
3 ft is half full.
a. How much work is performed in pumping all the water over the top edge of
the tank?
b. How much work is performed in draining all the water from the tank through
a hole at the tip?
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Modeling Fluid Pressure and Force
Water pressure increases with depth.
Pascal’s principle: any point in a liquid the pressure is the same in all
directions.
If a plate with surface area A is submerged
horizontally at a depth h in a fluid, the
force exerted by the fluid on the surface
of the plate is
F = PA = (δh)A = ρghA
where
P = δh is the pressure,
δ = ρg is the weight density,
ρ is the mass density,
g is the acceleration due to gravity.
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Fluid Force
If a plate is submerged vertically in a fluid of a weight density δ = ρg (lb/ft3 ) and
that the submerged portion of the plate extends from h = a to h = b on the vertical
axis.
Then the total force F exerted by the
fluid is given by
Z
F =
b
δhL(h)dh
(20)
a
where
h is the depth
L(h) is the corresponding length of
a typical horizontal approximating
strip.
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Example 31 (Fluid force on a vertical surface)
The cross sections of a certain trough are inverted isosceles triangle with height 6
ft and base 4 ft as shown in figure below. Suppose the trough contains water to a
depth of 3 ft. Find the total fluid force on one end.
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Example 32 (Modeling the force on one face of a dam)
A resevoir is filled with water to the top of a dam. If the dam is in the shape of a
parabola 40 ft high and 20 ft wide at the top, as shown in figure below, what is
the total fluid force on the face of the dam?
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Moments and Centers of Mass
Consider a thin plate (lamina) of uniform density ρ that covers the region R
bounded by the curves y = f (x) and y = g (x) in the interval [a, b].
The mass of R is
Z
m =(density) × (area) = ρ
b
[f (x) − g (x)] dx
a
The the moments of R about the y -axis
and the x-axis is given by
Z
b
My = (mass) × (distance to the y -axis) = ρ
x [f (x) − g (x)] dx
a
Z
Mx = (mass) × (distance to the x-axis) = ρ
a
b
o
1n
2
2
[f (x)] − [g (x)] dx
2
The center of mass (x̄, ȳ ) of the plate R is defined so that
mx̄ = My
and mȳ = Mx
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Example 33
(a) Find the center of mass of a semicircular plate of radius r .
(b) Find the centroid of the region bounded by the curves y = cos x, y = 0,
x = 0, and x = π/2.
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Example 34
A homogeneous lamina R has constant density ρ = 1 and is bounded by the
parabola y = x 2 and the line y = x. Find the mass and the centroid of R.
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Theorem of Pappus
The solid generated by revolving a region R about a line outside its boundary (but
in the same plane) has volume V = As, where A is the area of R and s is the
distance traveled by the centroid of R.
Example 35
A torus is formed by rotating a circle of radius r about a line in the plane of the
circle that is a distance R (R > r ) from the center of the circle. Find the volume
of the torus.
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