# Integrating wrt a function

Apteronotus
When we integrate a function $$f(t)$$ with respect to t, we are finding the area under the curve $$f$$. Intuitively, this is very clear.

What is the intuition behind integrating a function with respect to another function?
ex.
$$\int f(t)dg$$
where g is itself a function of t?

Gold Member
I'm not sure there is an easy intuitive answer, beyond interpreting it mathematically.
Since g is a function of t, you can apply the chain rule to give

$$\int f(t)d[g(t)]=\int f(t)\frac{d[g(t)]}{dt}dt=\int f(t)\.{g}(t) dt$$

So you are finding the area under another curve that equals the first function weighted by the slope of the second.

l'Hôpital
What if g is not differentiable? In fact, the most interesting case is when g is not even continuous. Look up the Riemann–Stieltjes integral

Apteronotus