Concept preview

What you will be able to do

1. 01Build a Riemann sum by slicing $[a,\,b]$ into $n$ equal pieces of width $\Delta x=\dfrac{b-a}{n}$, choosing a sample point $x_i$ in each, and writing the total in sigma notation as $\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$.
2. 02Explain why the limit $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$ produces one exact number — the definite integral $\displaystyle\int_a^b f(x)\,dx$ — and identify the roles of the $\int$ sign, the limits $a$ and $b$, and the $dx$.
3. 03Translate fluently in both directions between a limit of a Riemann sum and the definite-integral notation it stands for, reading off $f$, $a$, and $b$ from either form.

Suppose we want the area under $y=f(x)$ between $x=a$ and $x=b$, where $f$ is positive. Start by cutting the interval $[a,\,b]$ into $n$ equal subintervals. Each subinterval has the same width, which we call $\Delta x$. Because the total length $b-a$ is shared equally among $n$ pieces, $\Delta x=\dfrac{b-a}{n}$. On each subinterval we build a rectangle: its width is $\Delta x$ and its height is the value of $f$ at some chosen point $x_i$ inside that subinterval. The area of the $i$-th rectangle is therefore height times width, $f(x_i)\,\Delta x$, and the combined area of all $n$ rectangles approximates the area under the curve.