Foundations — Riemann sums — left, right, midpoint; formal definition of definite integral

Before you can read a single line of the main note, you must own every symbol it fires at you. This page builds each one from nothing — plain words, then a picture, then why the topic needs it. Each item leans on the one before, so read top to bottom.

0. The graph — what a curve actually is

Picture a horizontal number line (the $x$-axis) and a vertical one (the height). At each $x$ the function tells you how tall the curve stands. The whole subject is about the region trapped between this curve and the $x$-axis — see the shaded blueprint below.

Why the topic needs it: the "area under a curve" is meaningless until you know a curve is just "height as a function of position". That shaded region is the prize we are chasing. See also Area under a curve.

1. The interval $[a,b]$ — where we measure

Picture two vertical fence-posts on the $x$-axis: one at $x=a$, one at $x=b$. We only care about the strip of area between those posts.

Why the topic needs it: an area has to have edges. $[a,b]$ says exactly where the region starts and stops.

2. Subscripts $x_0,x_1,\dots ,x_n$ — labelling many points

Picture a row of fence-posts standing between $a$ and $b$. Instead of inventing new letters for each, we reuse $x$ and pin a numbered tag on it: post number $0$, post number $1$, up to post number $n$.

Why the topic needs it: we will plant many cutting points across $[a,b]$ and must refer to each individually. Subscripts are the naming system.

3. $n$ and the partition — slicing the interval

Look at the figure: the fence from $a$ to $b$ is sliced by upright cutting lines into $n$ vertical strips.

Why the topic needs it: one curvy region is hard; many thin strips are each almost a rectangle. The partition is the act of chopping. Linked idea: Continuity and Integrability.

4. $\Delta x$ — the width of a strip

Picture the horizontal distance between two neighbouring cutting lines — that base length is $\Delta x$. If the total fence is $b-a$ wide and you cut it into $n$ equal pieces, each piece is the total divided by $n$: that is $\frac{b-a}{n}$.

Why the topic needs it: a rectangle's area is width × height. $\Delta x$ is that width. Miss it and you're adding heights, not areas.

5. Sample point $x_i^{\ast }$ — where we read the height

We have three natural spots — left edge, right edge, or middle of the strip. The figure shows all three heights for the same strip.

Why the topic needs it: a rectangle needs a single height, but the curve's height varies across a strip. The sample point is our decision of which single height to trust.

6. Summation $\sum$ — adding many terms at once

Why the topic needs it: we have $n$ rectangles to total. Writing $f(x_1^{\ast })\Delta x+f(x_2^{\ast })\Delta x+\dots$ is unbearable; $\sum$ says it in one breath. Handy closed forms live in Summation formulas.

Now the whole Riemann sum decodes: $S_n={\sum }_{i=1}^n\underset{\text{height}}{\underbrace{f(x_i^{\ast })}}\underset{\text{width}}{\underbrace{\Delta x_i}}=\text{total area of all }n\text{ rectangles.}$

7. The limit $\underset{n\to \infty }{\lim }$ — squeezing to the exact answer

The figure shows the staircase tightening as $n$ doubles.

Why the topic needs it: the integral is defined as this limit — it's the exact area the rectangles can only approach. Background: Limits of sequences.

8. The integral sign ${\int }_a^{b}f(x)dx$ — the finished notation

${\int }_a^{b}f(x)dx={\lim }_{n\to \infty }{\sum }_{i=1}^{n}f(x_i^{\ast })\Delta x.$

Why the topic needs it: this is the destination — every earlier symbol funnels into it. Once you see $\int$ as "$\sum$ taken to the limit", the notation stops being magic. The shortcut to evaluating it lives in the Fundamental Theorem of Calculus; refinements are the Trapezoidal Rule and Simpson's Rule.

How the foundations feed the topic

Each box is exactly one symbol you just learned; the arrows are "you need the left one to understand the right one".

Equipment checklist

Cover the right side and test yourself — you're ready for the main note only when every line is automatic.