Numerical integration
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Transcript of Numerical integration
Numerical 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.
Integration
Integration
Generally we use formulae to determine the integral of a function:
F(x) can be found if its antiderivative, f(x) is known.
aFbFdxxfb
a
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.
Integration
Figure shows the area under a curve using the midpoints
Integration
There are various integration methods: Trapezoid, Simpson’s, etc.
We’ll be looking in detail at the Trapezoid and variants of the Simpson’s method.
Trapezoidal Rule
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).
Trapezoidal Rule
Instead the curve is approximated using a sequence of straight lines, “slanted” to match the curve.
fi
fi+1
Trapezoidal Rule
Clearly the area of one rectangular strip from xi to xi+1 is given by
Generally is used. h is the width of a strip.
iiii xxff 11 I
) x- (x ½ h i1i
1...
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.
n1-n2102 f 2f 2f 2f f I h
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.
)f 2f
2f 2f (fdx xf
n1-n
2102
x
x
n
0
h
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.
Trapezoidal Rule
As an exercise show that the approximation given by the trapezium rule gives 23.9944.
Simpson’s Rule
Simpson’s 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.
Simpson’s Rule
Diagram showing approximation using Simpson’s Rule.
Simpson’s Rule
There are two variations of the rule: Simpson’s 1/3 rule and Simpson’s 3/8 rule.
Simpson’s Rule
The formula for the Simpson’s 1/3,
n1-n32103
x
x
f 4f 4f 2f 4f fdx xfn
0
h
Simpson’s 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.
Simpson’s Rule
The formula for the Simpson’s 3/8,
n1-n321083
x
x
f 3f 2f 3f 3f fdx xfn
0
h
If the number of strips is divisible by three we can use the 3/8
rule.
Thank You