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
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.