Page 150 - Excel for Scientists and Engineers: Numerical Methods
P. 150
Chapter 7
Integration
The solution of scientific and engineering problems sometimes requires
integration of an expression. Symbolic integration involves the use of the
methods of calculus to yield a closed-form analytical expression: the indefinite
integral, or mathematical function F(x) whose derivative dy/dx is given. We will
not attempt to find the indefinite integral -Excel is not equipped to do symbolic
algebra - but instead find the area under the curve bounded by a function F(x)
and the x-axis. This area is the definite integral.
It may be difficult or even impossible to obtain an expression for the integral
of a particular function. But by using numerical methods we can always obtain a
value for the definite integral. The result of numeric integration is the area under
the curve, between specified limits, from x = a to x = b. The calculation will
involve a curve described either by a table of experimental x, y values or by a
function y = F(x).
This chapter provides methods for calculating the area under a curve that is
described by a table of x, y values on a worksheet or by a worksheet formula.
Some methods require evenly spaced x values, while for others the x values can
be irregularly spaced.
Area under a Curve
By "area under a curve" we mean the area bounded by a curve and the x-axis
(the line y = 0), between specified limits. The area can be positive if the curve
lies above the x-axis or negative if it is below.
Calculation of the area under a curve is sometimes referred to as quadrature,
since it involves subdividing the area under the curve into a number of "panels"
whose areas can be calculated. The sum of the areas of the panels will be an
approximation to the area under the curve. The three most common approaches
are the rectangle method, in which the panels are rectangles, the trapezoid
method, in which the panels are trapezoids and Simpson's method, which
approximates the curvature of the function. These methods require that we have
127
