The derivative half is complete. This is the second of the two operations — and it is introduced before the Fundamental Theorem on purpose, so she learns what an integral is before she learns the shortcut for computing one.

Good courses spend a painful week on Riemann sums for exactly this reason. Students hate it; it feels like arithmetic busywork when a shortcut is visibly coming. It's the correct pedagogical decision, and if her teacher rushes it, that's the gap to fill — because a student who thinks "integral = antiderivative" cannot make sense of anything in Units 8 or 9.

Leibniz's ∫ is an elongated S, for summa. The dx is the width of an infinitesimal slice. So ∫ f(x) dx reads literally: the sum of (height × width).

Once you see that, the notation stops being a symbol to memorize and becomes a description of the procedure. It also explains why the dx is not optional decoration — it's the width factor, and in Unit 6 substitution it will need to be converted like any other quantity.

Area is the picture, not the idea. The integral accumulates a product where one factor won't hold still.

• rate × time = distance — but only if the rate is constant. If it varies, integrate.
• force × distance = work — only if force is constant. If it varies, integrate.
• density × volume = mass. Varying density? Integrate.
• probability density × interval = probability.
• load × distance = passenger-miles; queue length × time = total delay.

Every one of those is "multiply two things where one of them varies." Teaching integration purely as area is why students later can't recognize an integral when it shows up as work or as expected value. The unit on the integrand is multiplied by the unit on dx — always check it.

A trapezoid connects the two endpoints with a straight chord. On a concave-up curve the chord lies above the curve, so the trapezoid includes extra area → overestimate.

The midpoint rectangle is subtler. Its top is a horizontal line at the midpoint height — but tilt that line to be tangent at the midpoint and the area doesn't change, because the triangle you add on one side exactly matches the one you remove on the other. And on a concave-up curve the tangent lies below the curve → underestimate.

That's a genuinely nice argument and it's the reason midpoint is more accurate than trapezoid (about twice as accurate), which surprises people who expect the fancier-looking method to win.

The average of n numbers is their sum divided by n. For a continuous function there are infinitely many values, so "sum" becomes an integral and "divide by how many" becomes divide by the length of the interval.

Rearranged: ∫_a^b f = f_avg·(b−a). That's a rectangle of height f_avg and width (b−a) with the same area as the region — the flat level the curve would need if you smoothed it out.

Unit 2 left a hole: nothing in the power rule produces 1/x, because the rule always lowers the exponent by one and you'd have to start at x⁰. Here's the resolution. Define:

Then everything about logarithms falls out of geometry:

• d/dx[ln x] = 1/x is immediate from the Fundamental Theorem (Unit 6) — no longer a mystery entry in a table.
• ln(ab) = ln a + ln b comes from the substitution t → at: the area from 1 to ab splits into the area from 1 to a plus the area from a to ab, and scaling maps that second piece exactly onto the area from 1 to b. The log law is a scaling symmetry of the hyperbola.
• e is simply the number where the accumulated area first reaches 1.

Napier invented logarithms in 1614 as a pure calculating aid — a way to turn multiplication into addition for astronomers. That they are also a fact about the area under y = 1/x was discovered by Grégoire de Saint-Vincent in the 1640s, and it genuinely stunned people. Two completely unrelated-looking things turned out to be one thing.

Everything so far has been construction. This is the payoff, and it is the centre of the subject. Two problems with no visible relationship — find the slope of a tangent and find the area under a curve — turn out to be inverse operations.

Nobody saw this for two thousand years. Archimedes computed the area under a parabola in 250 BC. Fermat and Descartes had tangent methods by the 1630s. Barrow, Newton's own teacher, had a geometric version of the connection around 1660 and did not recognize what he was holding. Newton and Leibniz did, independently, within about a decade of each other.

Say this to her explicitly. It's the most surprising thing she'll learn all year, and it is routinely taught as a computational rule in forty minutes.

Before: every area problem was a separate feat of genius. Archimedes needed one brilliant construction for the parabola, and it told you nothing about the hyperbola.

After: find an antiderivative, subtract at the endpoints. An entire class of genius-required problems collapsed into a procedure. That's the actual revolution — not new answers, but the mechanization of a problem type.

It's also why the subject is called calculus, which just means "a small pebble used for counting." The name advertises that it's a method of reckoning, not a body of results.

Unit 5's MVT proved: two functions with the same derivative differ by a constant. So once you have one antiderivative F, every other one is F + C, and there are no others hiding.

That's what makes FTC Part 1 well-defined. If you and I pick different antiderivatives, we differ by a constant — and when we subtract F(b) − F(a), the constant cancels. We get the same answer. Without MVT there'd be no guarantee of that, and the shortcut wouldn't be reliable.

FTC Part 2 guarantees every continuous function has an antiderivative: ∫_a^xf is one, and it exists whether or not you can write it down.

But most such antiderivatives cannot be expressed in elementary terms. e^−x² is the famous case. Liouville proved in the 1830s that no elementary antiderivative exists — not "nobody has found one," but there is none. Same for sin(x²), for (sin x)/x, for e^x/x.

And yet ∫_−∞^∞ e^−x²dx = √π, exactly. The bell curve — the most consequential function in all of statistics — cannot be integrated by any technique in her course, but its total area is a clean closed form. (The trick: square the integral, convert to polar coordinates. That's Calc III, but it's a fifteen-minute story.)

The deep asymmetry: differentiation is an algorithm — feed in any elementary function, turn the crank, get an elementary answer, always. Integration is not. It's pattern recognition and luck. That is why Unit 6 is a bag of tricks rather than a method, and it's worth telling her before she starts wondering what's wrong with her.

FTC reduced integration to antidifferentiation. But antidifferentiation has no algorithm — that's the asymmetry flagged at the end of Unit 6. So what follows is a bag of techniques, ordered roughly by how often each one works.

AB stops at u-substitution. Everything after it is BC-only, which is why the course accelerates here. Two further notes on scope: partial fractions is on the AP but only for non-repeating linear factors, and trigonometric substitution is not on the AP at all — many teachers cover it anyway.

Each technique is a differentiation rule run backwards. Recognizing which rule is being reversed is how you choose.

The chain rule says d/dx[F(g(x))] = F′(g(x))·g′(x). Read that equation right to left: anything of the form (function of g) times (derivative of g) is the derivative of a composite, so its antiderivative is that composite.

What to look for: a function and its own derivative both present in the integrand, up to a constant factor. The substitution just gives you a bookkeeping system for exploiting that.

It doesn't solve the integral. It trades one integral for another, and the whole art is choosing u so the trade is favourable — meaning ∫v du is easier than ∫u dv.

Rule of thumb: pick u to be the thing that gets simpler when differentiated. ln x becomes 1/x (much simpler). x becomes 1 (simpler). sin x becomes cos x (no simpler). e^x stays e^x (no simpler).

The standard mnemonic for the priority order of u is LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Whichever appears earliest in that list, make it u. It's a heuristic, not a theorem, and it works most of the time.

Every polynomial with real coefficients factors into linear and irreducible-quadratic pieces. That's the Fundamental Theorem of Algebra (Gauss, 1799) doing hidden work — a theorem from a completely different branch of mathematics is what makes this integration technique reliable.

And every resulting piece integrates to a log or an arctan. So the whole class of rational functions is integrable in elementary terms, always. That's rare and worth noticing: it's one of the few places where integration does have an algorithm.

The boundary case p = 1 diverges. That razor-thin threshold — 1/x fails, 1/x^1.0001 succeeds — is the same threshold that will govern p-series convergence in Unit 10, and for the same reason. The integral test makes the connection explicit.

It's the growth hierarchy from Unit 1 again, deciding whether a tail shrinks fast enough.

Rotate y = 1/x for x ≥ 1 around the x-axis.

• Volume = π∫₁^∞x⁻²dx = π. Finite.
• Surface area ≥ 2π∫₁^∞x⁻¹dx = ∞. Infinite.

A solid you could fill with π cubic units of paint but could never finish painting. Torricelli found this in 1641 and it caused a real crisis — Hobbes thought it discredited the whole enterprise.

The resolution (mathematical surfaces have no thickness; real paint does) is less interesting than the mechanism: the two integrals sit on opposite sides of the p = 1 threshold. It's a physical illustration of a convergence boundary.

Next stop — vector calculus. Green's theorem, Stokes' theorem, and the Divergence theorem each say a version of "the integral over a boundary equals the integral of a derivative over the interior." They are all special cases of the generalized Stokes' theorem, ∫_∂Ωω = ∫_Ωdω.

Then — differential forms and differential geometry, where that statement is the definition rather than a theorem, and then general relativity, which is differential geometry with physics attached.