integral

Format: integral( f(x), x, x1, x2, samples )

Arguments: (num) f(x) Function or expression to be integrated

(node) x Name of the input variable in f(x)

(num) x1 Lower limit

(num) x2 Upper limit

(int) samples Optional minimum number of sample points; default = 8

Returns: (num) The definite integral of f(x) in the range x=x1 to x2

Description: Integral is equivalent to the standard mathematical expression

This primitive approximates the integral of f(x) by sampling it at various points between x1 and x2. An adaptive algorithm is used that concentrates samples in areas where convergence is slowest. This process minimizes the number of samples needed and allows integral to handle functions with discontinuities in the range of integration; e.g., a step function. Integral continues adding samples until the integral is approximated to the desired level of precision as set by the precision primitive. Because it is impossible to know how precise a result is without knowing the exact solution, integral stops converging when it is LIKELY that the result is within the desired precision range.

Either or both limits of integration can be set to infinity using the infinity primitive; e.g., integral(Normal(x),x,-infinity,infinity). In these cases, the function being integrated is assumed to converging to zero at rate of at least 1/x^2 as x approaches infinity.

The function being integrated does not have to be defined at either the lower or upper limit of integration. For example, integral will properly integrate the function sin(x)/x in the range 0 to 1 even though sin(0)/0 is not defined. The function must be defined throughout the interior range of integration.

It is possible in rare cases for integral to miss important features of a function if the function contains discontinuities, such as a pulse, or rapid oscillations. If integral gives you an answer that does not seem reasonable, you can force integral to use a finer mesh of sample points by providing a value for samples. Samples should be a power of 2; e.g., 8, 16, 32, 64, ... This argument determines the minimum number of equally spaced samples that will be used. It is likely that integral will generate many more samples in areas of slow convergence.

Note: If the expression f(x) is a tree node rather than a function based on x, the reset primitive must be used to cause the tree to recalculate on each iteration. Constants retain their value from the first evaluation and return this value on all subsequent evaluations. To cause the constant to be reevaluated, use an expression similar to

integral({reset,f},x,x1,x2)

rather than

integral(f,x,x1,x2)

Examples: integral(sin(x),x,0,PI) = 2

f(x):=x^2

integral(f(x),x,0,3) = 9

integral(1/x^2,x,1,infinity) = 1

See Also: derivative, precision, infinity