Alright, in New Math, the integral is easy and a breeze.
We have the concept of cell and we have the graph of the function with its derivative a vector that connects successive points of the graph. So all we have to do is calculate the area covered by the function in each specific cell.
For example, here is a picture of a cell for the function y = x in 10 Grid of the successive x-axis points of 2.2 to 2.3, note, not drawn to scale. || || /| || || ||
Now the two x-axis points of the cell are 2.2 and 2.3 and there is nothing but empty space in between. Now the picketfence that forms inside the cell with its triangle top a rectangle, the rectangle is 0.1 width and 2.2 length, and the triangle width is 0.1 and 0.1 height. The hypotenuse of the triangle is the vector and is the derivative. The arrow of the vector points upward to indicate it is +1 direction. The Complementarity function of y = x is a vector direction of -1, as is the function y = 1/x is -1.
In the picture, the cell goes from 0 all the way to 10 but our interest stops at the triangle atop the rectangle of the picketfence.
Now for the integral, it is easy to compute the area in each cell. Each cell will have either a triangle to compute the area, or a rectangle to compute the area or a combination of both called a picketfence to compute the area.
Each cell in mathematics is transparent and clear and one can zoom in on any and every cell. So the integration involves simply the additive sum of the areas inside each cell.
Now in Old Math, they could never have a cell concept nor transparency nor able to zoom in, because they never were smart enough to have a finite to infinity borderline and then deluded people and conned people into what they call a limit concept. The limit concept is just smoke and mirrors. And because the limit is just sheer fakery, it blossomed all these other fakeries of a Riemann integral, a Lebesgue integral and hundreds of other pathetic worthless integrals. When you have true math, you have just one type of integral that you can inspect each individual cell and see what is going on with both the derivative and the integral.
Now I must mention one special case of cells. Sometimes in a Grid system we do not stop with the last and largest finite number such as what 10 is for 10 Grid or what 1*10^603 is for the 10^603 Grid. Sometimes we have to go to the first macroinfinity number which in 10 Grid is 10.1. In the case of the function y = 1/x, when x=0 that was undefined in Old Math for division by 0 was undefined. That situation alters in New Math. In New Math we define division by 0 to be a infinity number. Now in the case of y = 1/x the infinity number most appropriate is 10.1, so for the function y = 1/x when x=0 then y = 10.1 and we have a tiny triangle inside the first cell which is a bit more area since the derivative vector connects to x=0.1 and y = 10.
Now division by 0 may sometimes call forth a infinity number that is in between two finite numbers rather than being the macroinfinity first infinity number. An example of this is where the function graph is near x=0 and x=0.1 and so the division by 0 would have a y = 0.01 or a 0.001.
In New Math, there are no discontinuous functions. In New Math, all functions are continuous, differentiable and integrable. -- More than 90 percent of AP's posts are missing in the Google newsgroups author search archive from May 2012 to May 2013. Drexel University's Math Forum has done a far better job and many of those missing Google posts can be seen here: