Integration

1
Integration
2
Integration

• Problem
• An experiment has measured the number of
  particles entering a counter per unit time
  dN(t)/dt, as a function of time. The problem may
  be to determine the number of particles that
  entered in the first second.

3
Integration

• Problem
• An experiment has measured the number of
  particles entering a counter per unit time
  dN(t)/dt, as a function of time. The problem may
  be to determine the number of particles that
  entered in the first second.

4
Integration

• Integration is an important in Physics.
• used to determine the rate of growth in bacteria
  or to find the distance given the velocity (s
  ?vdt) as well as many other uses.
• The most familiar practical (probably the 1st
  usage) use of integration is to calculate the
  area.

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Integration

• Generally we use formulae to determine the
  integral of a function
• F(x) can be found if its antiderivative, f(x) is
  known.

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Integration

• when the antiderivative is unknown we are
  required to determine f(x) numerically.

7
Integration

• when the antiderivative is unknown we are
  required to determine f(x) numerically.
• To determine the definite integral we find the
  area between the curve and the x-axis.
• This is the principle of numerical integration.

8
Integration

• The traditional way to find the area is to divide
  the area into boxes and count the number of
  boxes or quadrilaterals.

9
Integration

• One simple way to find the area is to integrate
  using midpoints.

10
Integration
Figure shows the area under a curve using the
midpoints
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Integration

• One simple way to find the area is to integrate
  using midpoints.
• The midpoint rule uses a Riemann sum where the
  subinterval representatives are the midpoints of
  the subintervals.
• For some functions it may be easy to choose a
  partition that more closely approximates the
  definite integral using midpoints.

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Integration

• The integral of the function is approximated by a
  summation of the strips or boxes.
• where

13
Integration

• Practically this is dividing the interval (a, b)
  into vertical strips and adding the area of these
  strips.

Figure shows the area under a curve using the
midpoints
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Integration

• The width of the strips is often made equal but
  this is not always required.

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Integration

• There are various integration methods Trapezoid,
  Simpsons, Milne, Gaussian Quadrature for
  example.
• Well be looking in detail at the Trapezoid and
  variants of the Simpsons method.

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Trapezoidal Rule
17
Trapezoidal Rule

• is an improvement on the midpoint implementation.

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Trapezoidal Rule

• is an improvement on the midpoint implementation.
  
• the midpoints is inaccurate in that there are
  pieces of the boxes above and below the curve
  (over and under estimates).

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Trapezoidal Rule

• Instead the curve is approximated using a
  sequence of straight lines, slanted to match
  the curve.

fi1
fi
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Trapezoidal Rule

• By doing this we approximate the curve by a
  polynomial of degree-1.

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Trapezoidal Rule

• Clearly the area of one rectangular strip from xi
  to xi1 is given by

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Trapezoidal Rule

• Clearly the area of one rectangular strip from xi
  to xi1 is given by
• Generally is used. h is
  the width of a strip.

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Trapezoidal Rule

• The composite Trapezium rule is obtained by
  applying the equation .1 over all the intervals
  of interest.

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Trapezoidal Rule

• The composite Trapezium rule is obtained by
  applying the equation .1 over all the intervals
  of interest.
• Thus,
• ,if the interval h is the same for each
  strip.

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Trapezoidal Rule

• Note that each internal point is counted and
  therefore has a weight h, while end points are
  counted once and have a weight of h/2.

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Trapezoidal Rule

• Given the data in the following table use the
  trapezoid rule to estimate the integral from x
  1.8 to x 3.4. The data in the table are for ex
  and the true value is 23.9144.

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Trapezoidal Rule

• As an exercise show that the approximation given
  by the trapezium rule gives 23.9944.

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Simpsons Rule
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Simpsons Rule

• The midpoint rule was first improved upon by the
  trapezium rule.

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Simpsons Rule

• The midpoint rule was first improved upon by the
  trapezium rule.
• A further improvement is the Simpson's rule.

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Simpsons Rule

• The midpoint rule was first improved upon by the
  trapezium rule.
• A further improvement is the Simpson's rule.
• Instead of approximating the curve by a straight
  line, we approximate it by a quadratic or cubic
  function.

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Simpsons Rule

• Diagram showing approximation using Simpsons
  Rule.

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Simpsons Rule

• There are two variations of the rule Simpsons
  1/3 rule and Simpsons 3/8 rule.

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Simpsons Rule

• The formula for the Simpsons 1/3,

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Simpsons Rule

• The integration is over pairs of intervals and
  requires that total number of intervals be even
  of the total number of points N be odd.

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Simpsons Rule

• The formula for the Simpsons 3/8,

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Simpsons Rule

• If the number of strips is divisible by three we
  can use the 3/8 rule.

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Simpsons Rule

• http//metric.ma.ic.ac.uk/integration/techniques/d
  efinite/numerical-methods/exploration/index.html