Applications of the Integral

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Chapter 5

• Applications of the Integral

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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.

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Find the area enclosed by the functions
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5.2 Volumes of Solids Slabs Disks Washers
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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
  the interval.

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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.

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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)

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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.

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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 lttltb where f and g exist and are
  continuous on ab and f(t) and g(t) are not
  simultaneously zero on (ab).
• If the curve is smooth we can find its length.

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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 xf(t)
  and yg(t) is

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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.

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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

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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

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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.

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Finding the center of mass let M moment m
mass sigma density
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Centroid For a planar region the center of
pass of a homogeneous lamina is the centroid.

• Pappuss 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.

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5.7 Probability and Random Variables

• Expectation of a random variable If X is a
  random variable with a given probability
  distribution p(Xx) then the expectation of X
  denoted E(X) also called the mean of X and
  denoted as mu is

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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

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Theorem A

• Let X be a continuous random variable taking on
  values in the interval AB 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(altXltb) F(b) F(a)

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