# Exercise 10 : Integral Calculation.

## Estimation of δ using the classical Monte Carlo

### Uniform + Normal distributions

```{r}
n <- 10e4
X <- runif(n, 0, 5)
Y <- rnorm(n, 0, sqrt(2))
Y[Y <= 2] <- 0

h <- function(x, y) {
  5 *
    sqrt(pi) *
    sqrt(x + y) *
    sin(y^4) *
    exp(-3 * x / 2)
}

I1 <- mean(h(X, Y))
I1
```

### Exponential + Normal distributions

```{r}
n <- 10e4
X <- runif(n, 0, 5)
Y <- rexp(n, 3 / 2)
Y[Y <= 2] <- 0
X[X <= 5] <- 0

h <- function(x, y) {
  4 / 3 * sqrt(pi) * sqrt(x + y) * sin(y^4)
}

I2 <- mean(h(X, Y))
I2
```