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INTEGRATION

Integration is one of the major calculus applications applied by engineers. Consider the integrated quantity, F, computed over a known line, s, from A to B of the function, f(s).
(1) F = B⌠⌡Af(s)ds
One of the most common applications is to compute Integrated quantities that define the performance of vehicles or components. This can include lift, drag, pitching moment, thrust, roll, etc.
In calculus and in many engineering courses, the focus is on continuous integration, which is a closed form solution. An typical example is
(2) F = 2π⌠⌡0sin(s)ds
However, most engineering integrations do not involve a continuous function, but instead uses discrete data points located along the interval in question. An example of this is the lift coefficient of an airfoil, which is obtained by integrating the pressure coefficient along the airfoil surface. In this instance, the equation of the integration is
(3) cℓ = B⌠⌡Acp(s)ds
This process to integrate discrete points can also be applied to continuous equations by generating data at designated intervals to create a discrete data set for integration.
The most common integration processes use Newton-Cotes integration schemes or “rules”. The concept of the Newton-Cotes rules is based on a polynomial replacement of the integration:
(4) F = B⌠⌡Af(s)ds ≈ B⌠⌡Afn(s)ds
where the new approximate function, fn(s), is a polynomial of the
(5) fn(s) = a0 + a1s + a2s2 + … + ansn
The coefficients of the polynomial are represented here as a0, a1, a2, …, an and the order of the polynomial is n. These are combined in many instances with piecewise integration for more complex functions.

Rectangular Rules
This is the simplest integration rule, but when used appropriately can provide very accurate, fast results. Here, each points stands alone as the zeroth-order coefficient. Since this is a constant function, the integration is simple:
(6) F ≈ B⌠⌡Af(s)ds = f(s)(B − A)
where f(s) can be computed at some point (usually the midpoint) of the interval. The result of this integration is to apply a constant value, f(s), to the interval of integration, B − A. This generates a rectangle that describes the integral or “area under the curve”, hence the name Rectangular. The truncation error associated with this integration rule is
(7) E = (f′(s)(B − A)2)/(2)
This rule is applied in many instances where there is a single point located at the midpoint between A and B where s falls within s ∈ (A, B).

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