Transcript Chapter 5

Chapter 5
Applications of the Integral
5.1 Area of a Plane Region
• Definite integral from a to b is the area
contained between f(x) and the x-axis on
that interval.
• Area between two curves is found by 1)
determining where the 2 functions
intersect, 2) determining which function is
the greater function over that interval, and
3) evaluating the definite integral over the
interval of greater function minus lesser
function.
Find the area enclosed by the functions:
y1  2 x 2  4, y2   x  7
3
2x  4  x  7  2x  x  3  0  x 
,1
2
y1 (0)  4, y2 (0)  7  y2  y1
2
2
1
  x  7  (2 x
3
2
1
2
 4)dx    x  3 2 x dx
2
3
2
1
  2x

x

  3 x   5.2
2
 3
 3
3
2
2
5.2 Volumes of Solids: Slabs,
Disks, Washers
b
V   A( x)dx
a
Solids of Revolution: Disk Method
• A solid may be formed by revolving a curve
about an axis.
• The volume of this solid may be found by
considering the solid sliced into many, many
round disks.
• The area of each disk is the area of a circle.
Volume is found by integrating the area. The
radius of each circle is f(x) for each x value in
b
the interval.
V     ( f ( x)) dx
2
a
Washer method
• If the area between two curves is revolved
around an axis, a solid is created that is
“hollow” in the center.
• When slicing this solid, the sections created
are washers, not solid circles.
• The area of the smaller circle must be
subtracted from the area of the larger one.
b


V     ( f ( x)  g ( x) dx
a
2
2
5.3 Volumes of Solids of
Revolutions: Shells
• When an area between two curves is revolved
about an axis a solid is created.
• This solid could be considered as the sum of
many, many concentric cylinders.
• Volume is the integral of the area, in this case it
is the surface area of the cylinder, thus: r = x
and h = f(x)
b
V  2  x f ( x)dx
a
Does it matter which method to
use?
• Either method may work. Sketch a picture
of the function to determine which method
may be easier.
• If a specific method is requested, that
method should be implemented.
5.4 Length of a Plane Curve
• A plane curve is smooth if it is
determined by a pair of parametric
equations x = f(t) and y = g(t), a
<=t<=b, where f’ and g’ exist and are
continuous on [a,b], and f’(t) and g’(t)
are not simultaneously zero on (a,b).
• If the curve is smooth, we can find its
length.
Approximate curve length by the
sum of many, many line segments.
• To have the actual length, you would need
infinitely many line segments, each whose
length is found using the Pythagorean
theorem.
• The length of a smooth curve, defined as
x=f(t) and y=g(t) is
b
L   [ f ' (t )]  [ g ' (t )] dt
2
a
2
What if the function is not
parametric, but defined as y = f(x)?
• Infinitely many line segments still provide the
length. Again, use the Pythagorean formula
with horizontal component = x and vertical
component = dy/dx for every line segment.
b
L
a
2
 dy 
1    dx
 dx 
5.5 Work & Fluid Force
• Work = Force x Distance
• In many cases, the force is not constant
throughout the entire distance.
• To determine total work done, add all the
amounts of work done throughout the interval –
INTEGRATE!
• If the force is defined as F(x), then work is:
b
W   F ( x)dx
a
Fluid Force
• If a tank is filled to a depth h with a fluid of
density (sigma), then the force exerted by the
fluid on a horizontal rectangle of area A on the
bottom is equal to the weight of the column of
fluid that stands directly over that rectangle.
• Let sigma = density, h(x)=depth, w(x)=width,
then force is:
b
F    h( x) w( x) dx
a
5.6 Moments and Center of Mass
• The product of the mass m of a particle and its
directed distance from a point (its lever arm) is
called the moment of the particle with respect
to that point. It measures the tendency of the
mass to produce a rotation about the point.
• 2 masses along a line balance at a point if the
sum of their moments with respect to that point
is zero.
• The center of mass is the balance point.
Finding the center of mass:
let M = moment, m = mass,
sigma = density
b
_
M
x

m
x

(
x
)
dx

a
b

(
x
)
dx

a
Centroid: For a planar region, the
center of pass of a homogeneous
lamina is the centroid.
• Pappus’s Theorem: If a region R, lying on
one side of a line in its plane, is revolved
about that line, then the volume of the
resulting solid is equal to the area of R
multiplied by the distance traveled by its
centroid.
5.7 Probability and Random
Variables
• Expectation of a random variable: If X is a
random variable with a given probability
distribution, p(X=x), then the expectation of
X, denoted E(X), also called the mean of X
and denoted as mu, is:
n
  E ( x)   xi pi
i 1
Probability Density Function (PDF)
• If the outcomes are not finite (discrete), but
could be any real number in an interval, it is
continuous.
• Continuous random variables are studied
similarly to distribution of mass.
• The expected value (mean) of a continuous
random variable X is
B
  E ( X )   x f ( x) dx
A
Theorem A
• Let X be a continuous random variable
taking on values in the interval [A,B] and
having PDF f(x) and CDF (cumulative
distribution function) F(x). Then
• 1. F’(x) = f(x)
• 2. F(A) = 0 and F(B) = 1
• 3. P(a<=X<=b) = F(b) – F(a)