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Use simulation method to calculate the integrals

\(\int^1_0 \exp(-exp({x\over2}))\)

and

\(\int^1_0\int^1_0 \exp(-exp({xy}))dxdy\)

as accurately as you can

n = 1e6
x <- runif(n)
f <- function(x)exp(-exp(x/2))
integrate(f, 0,1)

## 0.2779746 with absolute error < 3.1e-15

library(pracma)

f <- function(x,y) {
 exp(-exp((x*y)))
}
integral2(f,0,1,0,1)

## $Q
## [1] 0.2800888
## 
## $error
## [1] 3.717861e-09