```
## Registered S3 methods overwritten by 'ggplot2':
## method from
## [.quosures rlang
## c.quosures rlang
## print.quosures rlang
```

Let’s do an integral. What is the total area under the curve \(f(x)=x^2\) from \(x=-1\) to \(x=2\) and from \(x=4\) to \(x=5\)? We typically write this as \(\int_{-1}^{2} x^2 \, dx + \int_{4}^{5} x^2 \, dx\). Remembering some elementary calculus, we have \[\int_{-1}^{2} x^2 \, dx + \int_{4}^{5} x^2 \,dx = \\\int_{-1}^{2} \frac{d}{dx} \frac{x^3}{3} \, dx + \int_{4}^{5} \frac{d}{dx} \frac{x^3}{3} = \\\left.\frac{x^3}{3}\right|_{-1}^{2} + \left.\frac{x^3}{3}\right|_{4}^{5} = \frac{70}{3}.\] So the answer to the question is \(\frac{70}{3}\) units of area.

In order to see what’s actually going on here, let’s write a function to plot the curve. Fortunately this is easy as `ggplot2`

package provides the `stat_function`

function which does most of the work for us.

```
plot_curve <- function(curve, domain) {
ggplot() +
stat_function(fun = curve, aes(domain, curve(domain))) +
geom_vline(xintercept = 0) +
geom_hline(yintercept = 0) +
scale_x_continuous(expand = c(0,0)) +
ylab("") + xlab("")
}
plot_curve(function(x) x^2, c(-1,5))
```

So pretty standard stuff. `ggplot::stat_function`

does most of the work here, sampling across the domain of the given function and evaluating for us.

```
plot_auc <- function(curve, domain, regions) {
samples <- Map(function(r) data.frame(xs = seq(min(r), max(r), length.out = 100)), regions)
geoms <- Map(function(s) geom_area(data = s, aes(x = xs, y = curve(xs)), alpha = .5), samples)
return(Reduce(`%+%`, geoms, init = plot_curve(curve, domain)))
}
plot_auc(curve = function(x) x^2, domain = c(-1,5), regions = list(c(-1,2), c(4,5)))
```

In general it is not easy to find antiderivatives such as above, so we use numerical integration schemes in order to approximate solutions. I will be back in future posts to discuss some methods for doing this.