I do understand that we define the integral of δ(t) equal to 1, i know that we purposely choose n and 1/n as shown in the picture below, but i need help please to improve my understanding over it. I know that if i use high school geometry Area= base x height = n x 1/n = 1, but that is not using the standard calculus numerical method.
I am scouring over other mathstack's threads and trying to understand δ(t) as much as possible. I try to numerically evaluate the integral and i cannot get it equal to 1. That is the essence of my question and my problem.
$$\begin{align}
Area &=\int_0^n \delta(x)dx\\
&= \int_0^n \frac1n\, dt \\
&=-[\frac{1}{n^2}]\Bigl\vert_0^n \\
&=-[\frac{1}{n^2}-\frac{1}{0^2}]\\
&= problem:\,cannot\, divide\, by\, zero
\end{align}
$$
Is it possible to calculate numerically the inegral using the above standard method? How? What is the correct solution? I can't make it equal to 1. I ended up getting something that is impossible to compute.
When you write $\int_0^n \frac 1n dt$ you are integrating with respect to $t$, not $n$, so you just get $$\int_0^n \frac 1n dt=\left.\frac tn \right|_0^n=\frac nn-\frac 0n=1$$
There are a couple small problems with your model of the delta "function". The first is that it should be even, so your rectangle should be centered on zero. The second is that you should at least think of it with a $\lim_{n \to 0}$ out front. In fact, any shape will do equally well as long as it is strongly peaked around zero, has area $1$, converges to $0$ for all $x \neq 0$ in the limit.