Riemann Sums, Summation Notation, and Definite Integral Notation

A Riemann sum estimates the area under a curve by slicing the interval from a to b into n rectangles, picking a height f(x_i) on each one, and adding up height times width: sum of f(x_i) times delta x. That sum is only an estimate because real curves bend inside each rectangle. Push n toward infinity and let every rectangle's width shrink toward zero, and the jagged stack of rectangles hugs the curve exactly. The leftover error vanishes, and what remains is the definite integral, written integral from a to b of f(x) dx. The integral sign itself is a historical S, for sum, stretched out to remind you it's the limit of a sum, not a single number of pieces.

In that notation, dx is not decoration. It is the limit of delta x, the rectangle width, as the partition gets infinitely fine. Drop it and you lose track of what's being multiplied by each height — the expression stops meaning anything. This is also why the variable inside matters: integral f(x) dx and integral f(t) dt describe the same number, because x and t are just labels for the slicing variable.

This limit looks exactly like the limit that defined the derivative back earlier in the course, where delta x shrank to zero to turn an average rate into an instantaneous one. Here delta x shrinks to zero to turn a finite sum into a continuous total. The partition doesn't even need equal widths — the only requirement for convergence is that the mesh, the widest subinterval, shrinks to zero as n grows, since a few fat rectangles ruin the approximation even if thousands of thin ones surround them. Unit 6 builds from here straight to the Fundamental Theorem, which replaces this limit with an antiderivative evaluation.