The following is an explanation of the three typical ways of thinking about using Riemann sums to approximate an integral, left-point, midpoint, and right-point approximations. The total area will become the summation of an increasing number of rectangles in the domain whose widht will decrease proportinately. For a few rectangles, the total area will be a simple sum of a few products - the products being the individual rectangles height times width. For many rectangles one should use Faulhaber's formula and limits, and the correct answer will follow.
with (Student [Calculus1]) :
f := x ---> x^3 (1)
Essentially, integration against one variable (as in dx for this case) can be thought of as determining the
area under the curve (bounded by the y=0 line). So what one can do is replace the area with rectangles,
which in this case are equal width for the 'n' rectangles chosen (here there are 2, 5, and then 10, as in the
first three parts of your ...
The following shows a direct calculation of the integral of a function using a few numbers of Riemann Sum rectangles as well as an infinite number of Riemann Sum rectangles. It is also apparent how this formulation is related to the integral.