If there is a function, (for example, a sine, as in the example), is there a simple way of creating a coordinate system in which the integrals of the function are split equally?
What I am looking to achieve is for the area to be split into equal parts, as in the below image. (not to scale, I drew the lines manually just to illustrate the concept).
Or, even more simply, is there a function f which gives me where I have to split the x axis, given that I want the function g to be split into n equal-area parts?
I'm considering developing my own (possibly needlessly overcomplicated) solution, but before that, I would like to know whether this is a known problem with an already existing solution.
What we actually want here, is to find a relationship between $x_i$ and $x_{i+1}$ so that the area between them is the area of the whole function divided by the number of how many equal divisions we want.
For the sine in the example, the total area is
$$\int_0^\pi sin(x)dx = 2$$
So, the area between $x_i$ and $x_{i+1}$ has to be $2/n$, where $n$ is the number of divisions.
This area is given by
$$\int_{x_i}^{x_{i+1}} sin(x)dx = -cos(x_{i+1}) + cos(x_i) $$
Therefore
$$cos(x_i) - cos(x_{i+1}) = \frac{2}{n}$$
Now we can express the next point in our new coordinate system based on the previous one:
$$x_{i+1} = arccos(cos(x_i)-\frac{2}{n})$$