Question

What is cumulative sum and how it is related to integral calculus?

Answer 1

Ah, good question. First of all I believe you are talking about Riemann Sums. IF you take the limit of this sum of the number of rectangles to infinity, then you have an exact area under the given curve. This is because when you take the above limit what you are doing is decreasing the change in the x direction infitestimally so that you may obtain an exact solution or numerical value. Namely:

\[A=\lim_{n \rightarrow \infty}\sum_{k=1}^{n}f(x _{k})\Delta x\]

This is how its related to integral calculus:

\[A=\lim_{n \rightarrow \infty}\sum_{k=1}^{n}f(x _{k})\Delta x=\int\limits_{b}^{a}f(x)dx\]

And then to evaluate the integral on the right use the Fundamental Theorem of Calculus:

\[\int\limits_{a}^{b}f(x)dx=F(b)-F(a) where F is an antiderivative of f'\]

Answer 2

you can use ~ for space.
(as far as information goes, everything is right)