What exactly is an integral? I understand it is the area under a curve but what does it pertain to?

Answer 1

The most common example in everyday life is perhaps the time integral of velocity is distance.
Every little bit of time dt, you go a distance dx = v dt.
If v happens to be constant the integral is easy
x = v t + Xi
where Xi is the x value at t = 0
That is just the area under a horizontal line.
BUT v(t) might be some more complicated function, for example
v(t) = Vi + a t
then
x = Xi + Vi t + (1/2) a t^2