Topic 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation: express a Riemann sum in summation notation and define the definite integral as its limit.

What this topic is asking

The College Board (Topic 6.3) makes the Riemann sum exact by taking a limit. You must write a Riemann sum in summation notation, recognize the definite integral as the limit of Riemann sums as the number of subintervals tends to infinity, and translate between the two notations.

From sum to integral

A worked translation

Why the limit makes it exact

A finite Riemann sum is only an approximation, with error from the rectangles not matching the curve. As $n \to \infty$, the subintervals shrink, the rectangles hug the curve ever more tightly, and the error vanishes for a continuous function. The limit of the Riemann sums is therefore the exact signed area, which is the definite integral. This limiting process is the rigorous definition; the Fundamental Theorem of Calculus (next topics) gives the shortcut for evaluating it without ever computing the limit by hand. Understanding the definition is still examined, especially the reverse problem of reading a limit-of-sum as an integral.

Reading limits backward into integrals

The most common exam task in this topic is the reverse translation: given a limit of a Riemann sum, write the equivalent definite integral. The recipe is to match $\Delta x$ to $\frac{b - a}{n}$ to recover the interval width, match the sample point $x_i = a + i\,\Delta x$ to recover the lower limit $a$, and read the function applied to $x_i$ as the integrand. Equal-width sums make this systematic. The lower limit is whatever constant is added to the $i$-dependent term, and the upper limit follows from the width. Practicing this both directions builds fluency with the notation the rest of Unit 6 depends on.

Why the choice of sample point stops mattering

In the limit defining the integral, the sample point within each subinterval, left endpoint, right endpoint, or midpoint, no longer affects the result for a continuous function. As $n \to \infty$ the subintervals shrink to zero width, so the difference between the largest and smallest function value on each subinterval vanishes, and every choice of sample point converges to the same number. This is why textbooks define the integral with right endpoints for convenience while the answer is independent of that convenience. For a finite sum the choice matters (left and right sums differ), but in the limit it does not, which is part of what makes the definite integral a single well-defined quantity rather than a family of approximations.

Connecting notation to the area picture

The integral symbol is built to evoke the summation it replaces. The elongated S of $\int$ stands for sum, the integrand $f(x)$ is the height of an infinitesimally thin strip, and $dx$ is that strip's vanishing width, so $f(x)\,dx$ is a strip's area and $\int_a^b f(x)\,dx$ adds all the strips from $a$ to $b$. Reading the notation this way keeps the meaning of the definite integral, the accumulated signed area, attached to the symbols, rather than treating $\int \ldots dx$ as an opaque instruction. It also clarifies why the limits of integration sit on the integral sign: they mark the start and end of the region whose strips you are summing.