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