double integral in R

Question

I'm wondering how to code that takes double integrals in R. I already referred two similar questions.

calculating double integrals in R quickly

double integration in R with additional argument

But I'm still confused how I can get my question from those answers. My question is following.

I would like to code this calculations in R.

From my hand and Wolfram alpha calculation, it becomes 16826.4. I know how to take a integral if both integrals are from exact numbers using adaptIntegrate(). But I'm not sure how to do in my case. Could you guys help me? Thank you so much in advance.

Answer 1

You can actually use the pracma::integral2() function in order to solve this integral, but some work to define it as dydx is needed:

f <- function(x,y) x+0.805
xmin <- 15; xmax<-50; 
ymin<-function(x) x; ymax<-50;
integral2(f,xmin,xmax,ymin,ymax)
$Q
[1] 16826.4

$error
[1] 1.818989e-12

This since:

More info here

Answer 2

The domain of integration is a simplex with vertices (15,15), (50,15) and (15,50). Use the SimplicialCubature package:

> library(SimplicialCubature)
> S = cbind(c(15,15),c(50,15),c(15,50))
> adaptIntegrateSimplex(function(v) v[1]+0.805, S)
$integral
[1] 16826.4

$estAbsError
[1] 1.68264e-08

Answer 3

Let me start with the code and then step through to explain it.

InnerFunc = function(x) { x + 0.805 }
InnerIntegral = function(y) { sapply(y, 
    function(z) { integrate(InnerFunc, 15, z)$value }) }
integrate(InnerIntegral , 15, 50)
16826.4 with absolute error < 1.9e-10

The first line is very easy. We just need the function f(x) = x + 0.805 to be able to compute the inner integral.

The second step is the only thing that is tricky. It seems natural to compute the inner integral with a simpler expression function(z) { integrate(InnerFunc, 15, z)$value } and just integrate it. The problem with that is that integrate expects a vectorized function. You should be able to give it a list of values and it will return a list of values. This simple form of the first integral just works for one value at a time. That is why we need the sapply so that we can pass in a list of values and get back a list of values (the first definite integral).

Once we have this vectorized function for the inner integral, we can just pass that to integrate to get the answer.

Later Simplification
While the above sapply method worked, it is more natural to use the function Vectorize like this.

InnerFunc = function(x) { x + 0.805 }
InnerIntegral = Vectorize(function(y) { integrate(InnerFunc, 15, y)$value})
integrate(InnerIntegral , 15, 50)
16826.4 with absolute error < 1.9e-10