integrate
Integration of OneDimensional Functions
Adaptive quadrature of functions of one variable over a finite or infinite interval.
Usage
integrate(f, lower, upper, ..., subdivisions = 100L, rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol, stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)
Arguments
 f
 an R function taking a numeric first argument and returning a numeric vector of the same length. Returning a nonfinite element will generate an error.
 lower, upper
 the limits of integration. Can be infinite.
 ...
 additional arguments to be passed to
f
.  subdivisions
 the maximum number of subintervals.
 rel.tol
 relative accuracy requested.
 abs.tol
 absolute accuracy requested.
 stop.on.error
 logical. If true (the default) an error stops the
function. If false some errors will give a result with a warning in
the
message
component.  keep.xy
 unused. For compatibility with S.
 aux
 unused. For compatibility with S.
Details
Note that arguments after ...
must be matched exactly.
If one or both limits are infinite, the infinite range is mapped onto a finite interval.
For a finite interval, globally adaptive interval subdivision is used in connection with extrapolation by Wynn's Epsilon algorithm, with the basic step being GaussKronrod quadrature.
rel.tol
cannot be less than max(50*.Machine$double.eps,
0.5e28)
if abs.tol <= 0<="" code="">.
Value

A list of class
 value
 the final estimate of the integral.
 abs.error
 estimate of the modulus of the absolute error.
 subdivisions
 the number of subintervals produced in the subdivision process.
 message
"OK"
or a character string giving the error message. call
 the matched call.
"integrate"
with components
Note
Like all numerical integration routines, these evaluate the function on a finite set of points. If the function is approximately constant (in particular, zero) over nearly all its range it is possible that the result and error estimate may be seriously wrong.
When integrating over infinite intervals do so explicitly, rather than just using a large number as the endpoint. This increases the chance of a correct answer  any function whose integral over an infinite interval is finite must be near zero for most of that interval.
For values at a finite set of points to be a fair reflection of the behaviour of the function elsewhere, the function needs to be wellbehaved, for example differentiable except perhaps for a small number of jumps or integrable singularities.
f
must accept a vector of inputs and produce a vector of function
evaluations at those points. The Vectorize
function
may be helpful to convert f
to this form.
Source
Based on QUADPACK routines dqags
and dqagi
by
R. Piessens and E. deDonckerKapenga, available from Netlib.
References
R. Piessens, E. deDonckerKapenga, C. Uberhuber, D. Kahaner (1983) Quadpack: a Subroutine Package for Automatic Integration; Springer Verlag.
Examples
library(stats)
integrate(dnorm, 1.96, 1.96)
integrate(dnorm, Inf, Inf)
## a slowlyconvergent integral
integrand < function(x) {1/((x+1)*sqrt(x))}
integrate(integrand, lower = 0, upper = Inf)
## don't do this if you really want the integral from 0 to Inf
integrate(integrand, lower = 0, upper = 10)
integrate(integrand, lower = 0, upper = 100000)
integrate(integrand, lower = 0, upper = 1000000, stop.on.error = FALSE)
## some functions do not handle vector input properly
f < function(x) 2.0
try(integrate(f, 0, 1))
integrate(Vectorize(f), 0, 1) ## correct
integrate(function(x) rep(2.0, length(x)), 0, 1) ## correct
## integrate can fail if misused
integrate(dnorm, 0, 2)
integrate(dnorm, 0, 20)
integrate(dnorm, 0, 200)
integrate(dnorm, 0, 2000)
integrate(dnorm, 0, 20000) ## fails on many systems
integrate(dnorm, 0, Inf) ## works