The Integral

Step 1 · Trapping the curve with step functions: upper and lower sums

Step 2 · Refinement, bounds, and Darboux integrability

Step 3 · Witnessing the gap collapse: $\int_0^1 x^2\,dx = \frac{1}{3}$

Note · Why are partitions enough? We construct the upper and lower step functions only through "partitions" — might we be missing better squeezing schemes?

Interactive component 1 · First encounter with the squeeze: in equal-subdivision mode, use the logarithmic slider to take $n$ from 1 to 500 and watch the upper and lower staircases trap $f(x)=x^2$ while the gap $U-L$ collapses; switch to "manual" mode to drag, insert, and delete division points, freely constructing non-equal partitions. The "stacked gaps" animation explains why the equal-subdivision gap is $\frac1n$. Write your conjecture into "My conjecture"; Step 3 reveals the target value.

Interactive component 3 · Signed area: drag the upper limit of integration $b$ and watch the definite integral of $f(x)=x^2-0.25$ turn from negative to positive. Below the $x$-axis the area is negative (blue); past the zero it becomes positive (red) and gradually cancels the negative area.

Interactive component 4 · Monotonicity under refinement: keep clicking "Add one cut" and watch $L$ monotonically nondecreasing, $U$ monotonically nonincreasing, and the gap monotonically nonincreasing. The history table lets you look back and compare.

Interactive component 5 · Cross-partition comparison: drag the division points of $P_1$ (blue) and $P_2$ (red) separately and observe $L(P_1) \le U(P_2)$. Click "Show common refinement" to see the full chain of inequalities.

Interactive component 6 · The birth of sup and inf: drag the slider to increase the partition count $N$; the blue dots (lower sums) approach the blue dashed line $\sup L$ and the red dots (upper sums) approach the red dashed line $\inf U$. When the gap can be made arbitrarily small, the two lines coincide at $I$.

Interactive component 7 · The $\varepsilon$-challenge: the system hands you an $\varepsilon$; you must adjust the subdivision count $n$ so the gap $U-L < \varepsilon$. Switch to the Dirichlet counterexample mode to experience non-integrability.

Interactive component 8 · The zero-distance criterion: shrink $\varepsilon$ and watch the blue shading centered at $A$ narrow. If $B$ always stays inside, $B$ is "squeezed" into coincidence with $A$. Switch to the integral-bridge mode to see $I$ and $1/3$ squeezed together.

Interactive component 9 · Isolating jump points: click "Isolate" to enclose the anomalous values at the jump points in small intervals; adjust $\delta$ and watch the gap $\to 0$, while the old integral value $S=8$ remains trapped by $L \le S \le U$.

Interactive component 10 · The despair of the Dirichlet function: whether $n$ is 1 or 500, every subinterval contains blue points (rationals) and red points (irrationals); the upper staircase stays at 1, the lower staircase stays at 0, and the gap sits motionless at 1.

Auxiliary tool 2 · Logic-direction indicator: Action one (integrability) belongs to the existential direction — producing one partition suffices; Action two (locking the value) belongs to the universal direction — $I$ is trapped by every partition.

Fill in: $f(x)=x^2$ is monotonically increasing on $[0,1]$. After equal subdivision into $n$ pieces, what are $M_k$ and $m_k$ on the $k$-th subinterval $[\frac{k-1}{n}, \frac{k}{n}]$?

Interactive component 11 · The square tower and the telescope: in the left panel stack the towers $1^2, 2^2, \dots, n^2$ to verify $\sum k^2 = n(n+1)(2n+1)/6$; in the right panel watch the telescoping cancellation of $(k+1)^3 - k^3$.

Auxiliary tool 1 · The $\varepsilon$-$N$ language translator: enter an $\varepsilon$, and the system automatically generates the corresponding $n$ (taking $n > 1/\varepsilon$) and verifies $1/n < \varepsilon$, breaking "$\forall \varepsilon, \exists n$" down into a game.

Interactive component 12 · Witnessing the gap collapse: drag $n$ from 1 to 500 (with a one-click ×10), with four numbers on screen $L_n, U_n, \frac1n, \frac13$. Enter an $\varepsilon$ to draw a marker line; when the gap dips below it, "integrability proved" fires.

Interactive component 13 · The squeeze of Action two: on the number line the interval $[L_n, U_n]$ narrows as $n$ grows, and the big question mark $I$ is forced into coincidence with $\frac13$.

Interactive component 14 · From area to integral, and back: three tabs — area axioms (Chapter 1), the definite integral (Chapter 2), and the closed loop (dashed lines linking the matching concepts). Drag $n$ and watch the gaps in both figures shrink in sync, finally closing at $1/3$.