chapter 9 numerical integration flow charts, loop structures copyright © the mcgraw-hill companies,...
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Chapter 9Numerical Integration
Flow Charts, Loop Structures
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Concepts From Calculus
• Two fundamental operations:– Differentiation– Integration
• Differentiation: finding the rate of change (derivative) relative to a variable
• Integration: summing rates of change over an interval
• We will illustrate with linear position, velocity, and acceleration of a particle
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Linear Motion
• Consider this particle (note: in mechanics, the term particle does not relate to size. A particle is simply a body for which we can neglect rotational motions. With this definition, a car or even a planet can be considered a particle).
• It moves along a straight line. Its position is measured from some reference point.
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x
Linear Motion
• As the particle moves a distance Δx during a time period Δt, we say that the average rate of change of the position is Δx /Δt
• The velocity, the instantaneous rate of change, is found by taking the limit as Δt approaches zero:
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xΔx
Linear Motion
• We say that the velocity is the derivative of position with respect to time
• Similarly, the acceleration of the particle is the instantaneous rate of change of velocity, or acceleration is the derivative of velocity with respect to time:
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Linear Motion
• Now let’s work the other way: if we know that our speed over a period of time, can we determine how far we have gone?
• Start with the equation for velocity:
• This can be rewritten as:
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Linear Motion
• So over a very small period of time dt, the change of position dx will be the velocity times dt
• To get the change of position over a larger time interval, we sum (integrate) v times dt for all of the small time intervals:
Engineering Computation: An Introduction Using MATLAB and Excel
Graphical Interpretations
• Consider position x plotted vs. time t• The change in x divided by the change in t is the
average rate of change over that time interval:
Engineering Computation: An Introduction Using MATLAB and Excel
Δt
Δx
Graphical Interpretations
• As the interval time becomes infinitesimally small, the rate of change becomes the slope of the curve at a specific point in time
Engineering Computation: An Introduction Using MATLAB and Excel
t
Slope = derivative of x with respect to t
Graphical Interpretations
• Now consider a graph of velocity vs time• The change in position over a small time interval
dt is v times dt
Engineering Computation: An Introduction Using MATLAB and Excel
dt
v
Graphical Interpretations
• The change in position is the sum of these small areas – the integral of velocity over the time interval
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t1 t2
Graphical Interpretations
• The derivative of a function is the slope of the curve of that function at a particular point
• The integral of a function is the area under the curve of that function for a given interval
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Derivatives of Polynomials
• In calculus classes, you have learned (or will learn) how to differentiate many types of functions
• Polynomials are simplest to differentiate: for each term, reduce the exponent of the variable by one, and modify the term by the original exponent
• Example:
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Integrals of Polynomials
• To integrate a polynomial, reverse the process: for each term, add one to the exponent of the variable and divide the term by the new exponent
Note the constant term C that must be added. This term will depend on a boundary condition
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Definite Integrals
• The previous integral, with no limits given, is called an indefinite integral, a general solution
• When upper and lower limits are defined, this is classed a definite integral
• To evaluate a definite integral, substitute the upper and lower limits into the general solution and subtract the value at the lower limit from that of the upper limit
Engineering Computation: An Introduction Using MATLAB and Excel
Definite Integrals
• Example:
• Note that the constant C was ignored, its value at the lower limit is the same as at the upper limit, and so is cancelled in the subtraction
Engineering Computation: An Introduction Using MATLAB and Excel
Numerical Integration
• We can get an approximate value of a definite integral of any function with a numerical approach
• The key is to remember that the value of the integral is simply the area under the function’s curve
Engineering Computation: An Introduction Using MATLAB and Excel
Numerical Integration
• Here is the area under the curve for our example, The total area is 16 (units depend on the units of x and y)
Engineering Computation: An Introduction Using MATLAB and Excel
Numerical Integration
• Consider the two values of x shown here. • To find the area under the curve between these
two points, we will approximate this portion of the curve with a straight line segment
Engineering Computation: An Introduction Using MATLAB and Excel
Numerical Integration
• A trapezoidal area is formed:
Engineering Computation: An Introduction Using MATLAB and Excel
Numerical Integration Example
• We will use four intervals of x, each one unit wide• Values of y at the interval endpoints:
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y = -3
y = 24
y = 9
y = 0y = 0
Numerical Integration Example
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• Areas:
• Sum = 18
A = -1.5 A = -1.5
A = 16.5
A = 4.5
Numerical Integration Example
Engineering Computation: An Introduction Using MATLAB and Excel
• Notice that our trapezoids over estimate the positive area and under estimate the negative areas
A = -1.5 A = -1.5
A = 16.5
A = 4.5
Numerical Integration Example
Engineering Computation: An Introduction Using MATLAB and Excel
• Using more intervals improves the accuracy:
• Sum = 16.5
A = -1.3125
A = -0.5625
A = 6.1875
A = 0.9375
A = 3.1875
A = 9.9375
A = -1.3125
A = -0.5625
Numerical Integration Example
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• How do we know when we have used enough intervals? (Assuming that we don’t know the exact answer, which is probably why we are using a numerical solution)
• Try more intervals until the solution converges to a value
• This makes a MATLAB solution a good choice – we can change the number of intervals easily