VitalSigns VitalSigns - 1 month ago 8
R Question

integrating a function with multiple variables

I am new to R and need some help using integration.
I have a function defined as:

a <- function(t) { exp(-r1*t) }


I have another function which uses this function and is defined as:

b <- function(t,x) { a(t-x)* exp(-(1-exp(-r2*x))) }


where,
r1
and
r2
are constants.

Now, I need to integrate the function
b(t,x)
for values of
x
that range from
0
to
t
; given that
x <= t
.

I am not sure how to proceed with this. I tried the following, but I am not sure how to tell R to integrate over 'x' and not 't'.

c <- integrate(b, lower=0, upper=10)


When i run this, I get an error saying:


Error in a(t -x) : argument "t" is missing, with no default


Thanks in advance,

-S

Answer

I am using r1 = r2 = 1 as an example.

Since you want a double integral:

\int_{t=0}^{t=10} \int_{x=0}^{x=t} b(x,t)

the most basic approach is to apply integrate() twice.

First, you need to define a vectorized function evaluating the inner integral:

z <- function(t) sapply(t, function (t_i) integrate(b, lower = 0, upper = t_i, t = t_i)$value)

We can check that:

z(1:3)
# [1] 0.4225104 0.4440731 0.4150334

Now, we apply the outer integral to z:

integrate(z, lower = 0, upper = 10)
# 3.795567 with absolute error < 6.2e-06

My answer just aims to give you starting point. The linked post: calculating double integrals in R quickly gives you better approach for doing double integral.