Engineering Mathematics I
B.Eng(Hons) in Aerospace Engineering
B.Eng(Hons) in Mechanical Engineering
Application of Integration
Begin by using rectangles to approximate the area between
two curves.
Application of Integration
If the limit as n → ∞ exists, we will get the exact area,
which we recognize as a definite integral.
AREA BETWEEN TWO CURVES
Application of Integration
WORKED EXAMPLE
Find the area bounded by the graphs of y = 3 − x and
y = x2 − 9. [Ans: 343/6]
Application of Integration
WORKED EXAMPLE
Find the area bounded by the graphs of y = x2 and
y = 2 − x2 for 0 ≤ x ≤ 2.
[Ans: 4]
Application of Integration
WORKED EXAMPLE
Find the area bounded by
the graphs of y = x2, y = 2 − x
and y = 0.
[Ans: 5/6]
Application of Integration
Application of Integration
WORKED EXAMPLE
Repeat previous example but
integrate with respect to y
instead.
[Ans: 5/6]
Application of Integration
Volumes by Slicing
Define a cylinder as any solid whose cross sections
perpendicular to some axis - running through the solid -
are all the same.
Volume = (cross sectional area) × (height)
Application of Integration
Volumes by Slicing
Application of Integration
Application of Integration
We use the same process followed here to derive many
important formulas.
In each case, we divide an object into n smaller pieces,
approximate the quantity of interest for each of the small
pieces, sum the approximations and then take a limit,
ultimately recognizing that we have derived a definite
integral.
For this reason, it is essential that you understand the
concept behind the previous formula.
Application of Integration
WORKED EXAMPLE
The Pyramid Arena in Memphis has a square base of side
approximately 600 feet and a height of approximately 320
feet. Find the volume of the pyramid with these
measurements. [Ans: 38 400 000 ft3]
Application of Integration
The Method of Disks
Suppose that f (x) ≥ 0 and f is continuous on the interval
[a, b]. Take the region bounded by the curve y = f (x) and
the x-axis, for a ≤ x ≤ b, and revolve it about the x-axis,
generating a solid.
Application of Integration
The Method of Disks
We can find the volume of this solid by
slicing it perpendicular to the x-axis and
recognizing that each cross section is a
circular disk of radius r = f (x).
We then have that the volume of the
solid is
Since the cross sections of such a solid of revolution are
all disks, we refer to this method of finding volume as the
method of disks.
Application of Integration
WORKED EXAMPLE
Revolve the region under the curve on the
interval [0, 4] about the x-axis and find the volume of the
resulting solid of revolution. [Ans: 8p]
Application of Integration
Application of Integration
When using the method of disks, the variable of
integration depends solely on the axis about which you
revolve the two-dimensional region:
• revolving about the x-axis requires integration with
respect to x
• revolving about the y-axis requires integration with
respect to y.
Application of Integration
The Method of Washers
One complication that occurs in
computing volumes is that the
solid may have a cavity or “hole”
in it.
Application of Integration
Another complication occurs
when a region is revolved about a
line other than the x-axis or the y-
axis.
Application of Integration
WORKED EXAMPLE
Let R be the region bounded by the graphs of y = x2/4, x =
0 and y = 1. Compute the volume of the solid formed
by revolving R about the x-axis. [Ans: 56p/15]
Application of Integration
WORKED EXAMPLE
Let R be the region bounded by y = 4 − x2 and y = 0. Find
the volume of the solid obtained by revolving R about the
line x = 3. [Ans: 64p]
Application of Integration
Finding the volume of the solid shown below by the
method of washers is awkward.
Application of Integration
But, note that revolving a rectangle about the y-axis
creates a thin cylindrical shell (a hollow cylinder or pipe).
Application of Integration
To find the volume of this thin cylindrical shell, imagine
cutting the cylinder from top to bottom and then flattening
out the shell.
Application of Integration
Cylindrical Shells
Application of Integration
the method of cylindrical shells
Application of Integration
WORKED EXAMPLE
Use the method of cylindrical shells to find the volume of the
solid formed by revolving the region bounded by the graphs
of y = x and y = x2 in the first quadrant about the y-axis.
[Ans: p/6]
Application of Integration
WORKED EXAMPLE
Find the volume of the solid formed by revolving the region
bounded by the graph of y = 4 − x2 and the x-axis about the
line x = 3. [Ans: 64p]
Application of Integration
WORKED EXAMPLE
Let R be the region bounded by the graphs of y = cos x and
y = x2. Compute the volume of the solid formed by
revolving R about the lines (a) x = 2 and (b) y = 2.
[Ans: 13.8; 10.1]
Application of Integration
VOLUME OF A SOLID OF REVOLUTION - checklist
• Sketch the region to be revolved and the axis of revolution.
• Determine the variable of integration (x if the region has a
well-defined top and bottom, y if the region has well-defined
left and right boundaries).
• Based on the axis of revolution and the variable of
integration, determine the method (disks or washers for x-
integration about a horizontal axis or y-integration about a
vertical axis, shells for x-integration about a vertical axis or y-
integration about a horizontal axis).
• Label your picture with the inner and outer radii for disks or
washers; label the radius and height for cylindrical shells.
• Set up the integral(s) and evaluate.
Application of Integration
Arc Length
How could we find the length of the portion of the sine
curve shown in the figure?
Application of Integration
Arc Length
Approximate the curve with several line segments joined
together.
As you would expect, the
approximation of length will get
closer to the actual length of
the curve, as the number of line
segments increases. This
general idea should sound
familiar.
Application of Integration
Arc Length
Assume that f is continuous on [a, b] and differentiable on
(a, b).
Begin by partitioning the interval [a, b] into n equal pieces:
Approximate the arc length si by the straight-line distance
between two points.
Application of Integration
Arc Length
By the Mean Value Theorem,
Application of Integration
As n gets large the approximation should provide the
exact arc length
Recognize this as the limit of a Riemann sum for
so that the arc length is given exactly by the definite
integral:
whenever the limit exists.
Application of Integration
WORKED EXAMPLE
Find the arc length of the portion of the curve y = sin x
with 0 ≤ x ≤ π. [Ans: 3.8]
Application of Integration
WORKED EXAMPLE
A cable is hung between two
poles of equal height that are 20
feet apart. A hanging cable
assumes the shape of a catenary,
the general form of which is
y = a cosh x/a = a/2 (ex/a + e−x/a).
In this case, suppose that the
cable takes the shape of
y = 5(ex/10 + e−x/10), −10 ≤ x ≤ 10.
How long is the cable? [Ans: 23.5 ft]
Application of Integration
When a linear segment is rotated about a line,
the resulting surface of rotation is the frustum
of a cone.
We take advantage of this observation to
develop a method for calculating surface
areas.
For the frustum of a cone shown, the curved
surface area is given by:
Application of Integration
For the problem of finding the curved surface area of a
surface of revolution, consider the case where f (x) ≥ 0 and
where f is continuous on the interval [a, b] and
differentiable on (a, b). If we revolve the graph of y = f (x)
about the x-axis on the interval
[a, b], we get a surface of
revolution.
Application of Integration
First partition the interval [a, b] into n pieces of equal size:
a = x0 < x1 < · · · < xn = b, where
for each i = 1, 2, . . . , n.
On each subinterval [xi−1, xi ], we
can approximate the curve by the
straight line segment joining the
points:
(xi−1, f (xi−1)) and (xi , f (xi )).
Application of Integration
Revolving this line segment around the x-axis generates
the frustum of a cone. The surface area of this frustum will
give us an approximation to the actual surface area on the
interval [xi−1, xi ].
The slant height of the frustum is
Application of Integration
Apply the Mean Value Theorem,
for some number ci ∈ (xi−1, xi ). This gives us
Application of Integration
The surface area Si of that portion of the surface on the
interval [xi−1, xi ] is approximately the surface area of the
frustum of the cone,
since if Δx is small,
Application of Integration
Repeating this argument for each subinterval [xi−1, xi ],
i = 1, 2, . . . , n, gives an approximation to the total surface
area S,
Application of Integration
The factor of
in the integrand corresponds to the arc length of a small
section of the curve y = f (x), while the factor 2π f (x)
corresponds to the circumference of the solid of revolution.
Application of Integration
WORKED EXAMPLE
Compute the surface area of the circle y = y4 for 0 x 1
that is revolved about the x axis
[Ans: 3.4]
Application of Integration
For any constant force F applied over a distance d, we
define the work W as W = Fd.
We extend this notion of work to the case of a nonconstant
force F(x) applied on the interval [a, b] as follows. First, we
partition the interval [a, b] into n equal subintervals, each
of width
and consider the work done on each subinterval.
Application of Integration
If Δx is small and F is continuous, then the force F(x)
applied on the subinterval [xi−1, xi ] can be approximated by
the constant force F(ci ) for some point ci ∈ [xi−1, xi ].
The work done moving the object along the subinterval is
then approximately F(ci )x. The total work W done is then
approximately
Application of Integration
Recognize this as a Riemann sum, which, as n gets larger,
approaches the actual work,
The more a spring is compressed (or
stretched) from its natural length, the
more force is required to further
compress (or stretch) the spring.
According to Hooke’s Law,
for some constant k (the spring
constant).
Application of Integration
WORKED EXAMPLE
A force of 3 pounds stretches a spring 1/4 foot from its
natural length.
Find the work done in stretching the spring 6 inches
beyond its natural length.
[Ans: 3/2 ft lb]
Application of Integration
WORKED EXAMPLE
A spherical tank of radius 10 feet is
filled with water. Find the work done
in pumping all of the water out
through the top of the tank.
Note: both the force and the
distance are nonconstant.
[Ans: 2.6 X 106 ft lb]
Application of Integration
Impulse J of a force F(t) applied over the time interval [a, b]
is defined to be
Using Newton’s second law and integrating:
(assuming the integral exists)
Impulse relates force and time to account for changes in
velocity.
Application of Integration
Center of Mass
To consider an object as a point-mass, treat it as a single
point, with all of the mass concentrated at that point.
The center of mass is the location at which we could
place the pivot of a seesaw and have the objects balance.
The denominator is the total mass of the system. The
numerator is called the first moment of the system.
Application of Integration
Center of Mass
More generally, for a system of n masses m1,m2, . . . ,mn,
located at x = x1, x2, . . . , xn, respectively, the center of
mass is given by the first moment (sum of all mass
moments) divided by the total mass,
The denominator is the total mass of the system. The
numerator is called the first moment of the system.
Application of Integration
Suppose that we wish to find the mass and center of mass
of an object of variable density ρ(x) (measured in units of
mass per unit length) that extends from x = a to x = b.
If the density varies throughout the object, we can
approximate the mass by partitioning the interval [a, b]
into n pieces of equal width
Application of Integration
On each subinterval [xi−1, xi ], the mass is approximately
ρ(ci)x, where ci is a point in the subinterval. The total mass
is then approximately
Recognize this as a Riemann sum, which approaches the
total mass as n→ ∞,
Application of Integration
WORKED EXAMPLE
A 30-inch baseball bat can be represented approximately
by an object extending from x = 0 to x = 30 inches, with
density ρ(x) = 32(1/46 + x/690)2 grams per inch.
This model of the density function takes into account the
fact that a baseball bat is similar to an elongated cone.
Find the mass of the object.
[Ans: 6.144 X 10– 2 slug = 3.1 oz]
Application of Integration
By dividing an object into intervals and recognizing the
sum of ciρ(ci )Δx as a Riemann sum, one can show that the
first moment of an object of variable density is:
Application of Integration
Using the expression for the first moment from the previous
slide, the center of mass of an object of variable density is
given by
Application of Integration
WORKED EXAMPLE
Find the center of mass of the baseball bat from previous
example
[Ans: 19.6 in]
Application of Integration
Consider a vertically oriented wall (a dam) holding back a
lake. It is convenient to
orient the x-axis vertically
with x = 0 located at the
surface of the water and the
bottom of the wall at
x = a > 0. In this way, x
measures the depth of a
section of the dam.
𝑎
F = PA = rgxA 𝐹= 𝑔 𝜌 𝑜 𝑥 𝑥𝑤 𝑥 𝑑𝑥
Application of Integration
WORKED EXAMPLE
A dam is shaped like a trapezoid with height 60 feet. The
width at the top is 100 feet and the width at the bottom is
40 feet.
Find the maximum
hydrostatic force
that the dam will
need to withstand.
[Ans: 66 044 160 lbs]
Questions ?
