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.

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.

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 1 st usage) use of integration is to calculate the area.

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

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

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

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

Integration Figure shows the area under a curve using the midpoints

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.

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

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

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

Integration There are various integration methods: Trapezoid, Simpson’s, Milne, Gaussian Quadrature for example. We’ll be looking in detail at the Trapezoid and variants of the Simpson’s method.

Trapezoidal Rule

is an improvement on the midpoint implementation.

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. fifi f i+1

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

Trapezoidal Rule Clearly the area of one rectangular strip from x i to x i+1 is given by

Trapezoidal Rule Clearly the area of one rectangular strip from x i to x i+1 is given by Generally is used. h is the width of a strip.

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

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.

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.

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 e x and the true value is

Trapezoidal Rule As an exercise show that the approximation given by the trapezium rule gives

Simpson’s Rule

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

Simpson’s Rule The midpoint rule was first improved upon by the trapezium rule. A further improvement is the 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,

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,

Simpson’s Rule If the number of strips is divisible by three we can use the 3/8 rule.

Simpson’s Rule ues/definite/numerical- methods/exploration/index.html#