# Confused by integration calc in my financial maths notes!

I have added a pic of the notes that I am looking at!

I’m a little confused by them.In integration calcs, I’m used to seeing the integral of $$f(x)$$ to get the difference between two points. Here it’s integrating $$f^\prime(x)$$ to get the difference.

I’m trying to picture the problem graphically - I’m assuming that $$M$$ is along the $$y$$-axis and $$t$$ is on the $$x$$-axis. In which case wouldn’t we just integrate $$M(t)$$? Unless $$M(t)$$ is supposed to be the CDF?

Basically I don’t understand why it’s the integral of $$M^\prime(t)$$ and not the integral of $$M(t)$$

Sorry getting very confused with this!

Any help is much appreciated • They're using this. In Wikipedia's notation $f=M',\,F=M$.
– J.G.
Jun 10 at 8:22

It's just fundamental theorem of calculus. We know that "$$M(t)$$ represents the total payment between timo $$0$$ and time $$t$$". Therefore it should be clear that, if you want the total payment between time $$t=\alpha$$ and $$t=\beta$$, you just have to calculate $$M(\beta)-M(\alpha)$$.
Now it's just a simple application of F.T.C.: it says that $$\int_a^bf(t)dt=F(b)-F(a)$$ where $$F(x)$$ is an antiderivative of $$f$$. Now replace $$f=g'$$. An antiderivative of $$g'$$ is $$g$$ obviously, and so you get $$\int_a^b g'(t)dt=g(b)-g(a)$$
Now plug in $$g(t)=M(t)$$ and you get your equation.