Riemann Sums, Summation Notation, and Definite Integral Notation AB & BC

Sigma notation packs a sum into four parts: an index that counts, a lower bound where the count starts, an inclusive upper bound where it stops, and a term evaluated at each value. A sum from $i = 2$ to $i = 5$ has four terms, since terms are one more than gaps. A Riemann sum on $[a, b]$ with $n$ equal pieces is $\sum_{i=1}^{n} f(x_i)\Delta x$ with $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$, so on $[1, 3]$ it reads $\sum_{i=1}^{n} f\!\left(1 + \frac{2i}{n}\right)\frac{2}{n}$, where the same $\frac{2}{n}$ is both the width of a rectangle and the spacing of the samples.

Letting $n \to \infty$ turns the sum into $\int_{a}^{b} f(x)\,dx$, and reading a given limit backwards is a match on three items: the factor playing $\Delta x$ gives the length of the interval, the constant sitting beside the $i$ term gives $a$, and the rest is $f$. The resulting symbol denotes a number. Its variable is a dummy, so $\int_{0}^{2} f(x)dx$ and $\int_{0}^{2} f(t)dt$ are equal; its limits are inputs to $f$ rather than values of $f$; and the $dx$ names the variable and carries the width that gives the answer its units.