Applications of Integration

1
Applications of Integration

• Volumes
• Arc Lengths
• Surface Areas
• Areas of Polar Domains

2
Summary of Formulae
Area bounded by the Graph of a Function
Area between the Graphs of Functions
Volume of a Solid of Revolution
Volume of the solid of revolution by letting the
domain between the graphs of the non-negative
functions f and g rotate about the x-axis.
Volume by Cylindrical Shells
3
Summary of Formulae
Volumes by Slicing
Arc Length
Length of a Parametric Curve
The Area of a Surface of Revolution
The Area of a Polar Domain
4
Overview of Problems 1/3
The bottom of a bowler is a disk of radius r
with center at the origin. Every intersection of
the bowler with a plane perpendicular to x axis
is a semicircle. Compute the volume of the
bowler.
1
Find the volume of the solid of revolution
obtained by letting the domain bounded by the
graph of the function f(x) x - x2 and the
x-axis rotate around the x-axis.
2
Find the volume of the solid of revolution
obtained by letting the domain in Problem 2
rotate around the y axis.
3
Find the volume of the solid of revolution
obtained by letting the disk bounded by the
circle (x-2)2 y2 1 rotate around the y axis.
4
Compute the volume of the cap of a ball x2y2z2
? r2 which lies above the plane zr-h.
5
6
Determine the volume of the intersection of the
cylinders x2z2 ? 1 and y2z2 ? 1.
5
Overview of Problems 2/3
7
Determine the length of the arc on the graph of
the function cosh(x) over the interval -1,1.
8
9
10
11
Find the area of the surface obtained by rotating
the curve y x3, 0 x 1, about the x-axis.
12
13
6
Overview of Problems 3/3
14
Use Maple to plot the polar curve r 1
cos(2?), 0 ? 2p. Find the area of the domain
bounded by this curve.
15
7
Computing Volumes
The bottom of a bowler is a disk of radius r
with center at the origin. Every intersection of
the bowler with a plane perpendicular to x axis
is a semicircle. Compute the volume of the
bowler.
Problem 1
This bowler is a half of a ball of radius r.
Hence the volume is half of the volume of a ball
of radius r.
Solution
To compute the volume of the bowler by
integration, slice it with planes perpendicular
to the x-axis as indicated in the picture.
The slice is a semicircle of radius (r2-x2)
1/2. The area of the slice is A(x)?(r2-x2)/2.
Conclude
8
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the domain bounded by the
graph of the function f(x) x - x2 and the
x-axis rotate around the x-axis.
Problem 2
The pictures below illustrates the curve and the
solid in question.
Solution
The volume can be computed by straightforward
application of the formula. The limits for
integration can be obtained by solving the
equation x x2 0.
To integrate expand the product
Conclude
9
Computing Volumes
Problem 3
Find the volume of the solid of revolution
obtained by letting the domain in Problem 2
rotate around the y axis.
Solution 1
We compute the volume in two
ways first we use the basic formula and then
Cylindrical Shells formula (much easier).
The pictures below illustrates the curve and the
solid in question.
To compute the volume use the standard formula
with the roles of x and y exchanged. Compute
first the volume of the solid obtained by letting
the domain bounded by the blue curve, the y-axis,
the x-axis and the line y1/4 rotate around the
y-axis. This solid is too big. We still have to
carve a niche corresponding to the red curve to
get the desired solid.
10
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the domain in Problem 2
rotate around the y axis.
Problem 3
Solution 1 (contd)
We compute the volume in two steps.
First we compute the volume V1 of the solid
obtained by letting the domain bounded by the
blue curve and the coordinate axes rotate around
the y-axis. From that we subtract the volume V2
corresponding to the red curve in the picture
i.e. we carve a niche in the first solid which
was too big.
11
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the domain in Problem 2
rotate around the y axis.
Problem 3
Solution 1 (contd)
We have now
To simplify expand the products in the integrals
and combine the integrals. One gets
12
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the domain in Problem 2
rotate around the y axis.
Problem 3
Using the Cylindrical Shells Formula the
computation is much easier
Solution 2 by Cylindrical Shells
13
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the disk bounded by the
circle (x-2)2 y2 1 rotate around the y axis.
Problem 4
Solution
The first task is to form a correct geometric
picture of the solid.
The disk pictured on the left rotates around the
vertical axis and forms the torus on the right
The volume can be computed in two ways. One can
adapt the method presented in the lecture on
Volumes or one can slice the solid in an
appropriate way. Here we will use the slicing
method.
14
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the disk bounded by the
circle (x-2)2 y2 1 rotate around the y axis.
Problem 4
Solution (contd)
Slice the torus by planes z t, -1 t 1.
The slice of the solid torus on the left is the
ring on the right.
Restricting to the xy plane, the plane z t
corresponds to the blue line on the left. The
portion of the blue line inside the red circle is
the width of the yellow ring above. Using this we
can compute the area of the slice.
15
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the disk bounded by the
circle (x-2)2 y2 1 rotate around the y axis.
Problem 4
Solution (contd)
Slice the torus by planes z t, -1 t 1.
The blue line is the intersection of the plane z
t in the xy plane.
16
Computing Volumes
Find the volume of the solid of revolution
obtained by letting the disk bounded by the
circle (x-2)2 y2 1 rotate around the y axis.
Problem 4
Solution (contd)
To get the volume V of the solid, simply
integrate the area of a general slice.
The above integral can also be computed by a
substitution, see the lecture on Volumes.
17
Computing Volumes
Compute the volume of the cap of a ball
x2y2z2 ? r2 which lies above the plane
zr-h.
Problem 5
Solution
The solid in question is the cap of the ball
above the red plane zr-h.
We slice the solid by planes parallel to the red
plane in the picture.
A slice by the plane z t is a disk of radius
(r2-t2)1/2. Hence the area of a slice is
?(r2-t2).
Conclude
18
Computing Volumes
Problem 6
Determine the volume of the intersection of the
cylinders x2z2 ? 1 and y2z2 ? 1.
The picture on the right illustrates the solid in
question.
The key to being able to compute the volume is to
slice the cylinders and the solid in question
properly.
19
Computing Volumes
Determine the volume of the intersection of the
cylinders x2z2 ? 1 and y2z2 ? 1.
Problem 6
Solution
Slice the solid by plane z t as illustrated in
the picture.
The slice of one of the cylinders by the red
plane is an infinite strip of width 2(1-t2)1/2.
The strips corresponding to the two cylinders are
perpendicular, and their intersection is a square
with side length 2(1-t2)1/2.
Hence the area of a slice is 4(1-t2).
Conclude
20
Computing Arc Lengths
Problem 7
Determine the length of the arc on the graph of
the function cosh(x) over the interval -1,1.
This is a straightforward application of the Arc
Length Formula.
Solution
.
21
Computing Arc Lengths
Problem 8
Solution
This is a straightforward application of the
formula. Intermediate simplifications are rather
technical, and one should use Maple there. We get
Use Maple to integrate.
22
Computing Arc Lengths
Problem 9
Solution
By the Fundamental Theorem of Calculus,
23
Computing Arc Lengths
Problem 10
Solution
One observes first that the curve is perfectly
symmetric with respect to both x- and the y-axis.
Using this observation the curve can be plotted
by Maple.
The expression for f was obtained by solving y
in terms of x form the equation of the curve.
Next use the Arc Length Formula to compute the
length of the blue arc. The length of the curve
is then simply four times the length of the blue
arc.
24
Computing Arc Lengths
Problem 10
Solution (contd)
The Arc Length Formula gives the following
integral for the length of the blue arc.
Simplifying the integrand (use Maple here) we get
Hence the total length of the above curve is
4x(3/2) 6
Observe that this is a converging improper
integral.
25
Surface Areas
Problem 11
Find the area of the surface obtained by rotating
the curve y x3, 0 x 1, about the x-axis.
Solution
This is a straightforward computation using the
formula for the area. We get
Use the substitution t 1 9x4.
26
Surface Areas
Problem 12
Solution
This is a straightforward computation using the
formula for the area. We get
Combine the square roots.
Use the substitution 2x2sinh2(t).
27
Polar Domains
Problem 13
Solution
This is a straightforward computation using the
formula for the area. We get
28
Polar Domains
Problem 14
Solution
It is useful to plot the curves in question
first.
Intersection points
This integral is the area bounded by the red
curve in the sector p/3? p/3 minus the area
bounded by the blue curve in the same sector.
29
Polar Domains
Problem 15
Use Maple to plot the polar curve r 1
cos(2?), 0 ? 2p. Find the area of the domain
bounded by this curve.
Solution
The Maple command plot(1cos(2t),t,t0..2Pi,co
ordspolar) gives the plot below
The polar curve r 1 cos(2?)
30
Formulae Areas and Volumes
Area between a curve and the x-axis
Area between the graphs of the functions f(x) and
g(x), axb
f
g
a
b
Volume of a solid of revolution obtained by
letting the domain between the graph of a
function f, axb, and the x-axis rotate about
the x-axis.
31
Formulae Volumes
Volume of the solid of revolution obtained by
letting the domain between the graphs of the
non-negative functions f(x) and g(x), a x b,
rotate about the x-axis.
The solid is bounded by the surfaces of
revolution shown in the picture.
f
Volume of a solid of revolution obtained by
letting a domain bounded by the graph of a
non-negative function f(x), 0 a x b,
rotate about the y-axis.
b
a
Volume by Cylindrical Shells
32
Formulae Arc Lengths
The length of the arc on the graph of a function
f, a x b
(x(t),y(t))
(x(t2),y(t2))
The length of the parametric curve (x(t),y(t)),
t1 t t2
(x(t1),y(t1))
33
Formulae the Area of a Surface of Revolution
The area of the surface of revolution obtained by
letting the graph of the function f(x), a x
b, rotate about the x-axis.
34
Formulae Area of Polar Domains
The area of a domain bounded by the polar curve r
r(?), and the half lines ??1, ??2