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What you will learn in Integral Calculus
1. Integration as Reverse Differentiation
Undoing a derivative, and the constant that undoing cannot recover.
2. Standard Integrals
Module 18's table of derivatives, read backwards — with one entry that has no partner.
3. Integration by Substitution
The chain rule reversed — and a pattern to recognise before it is a method to execute.
4. Integration by Parts
The product rule reversed, and a choice of u that makes it work or makes it worse.
5. Integration by Partial Fractions
Splitting a rational function into pieces each of which integrates to a logarithm.
6. Trigonometric Integrals
Identity first, then integrate — because the identity is what makes it possible.
7. Riemann Sums and Accumulation
Chop, approximate, add, then add more — and watch two estimates close on a single number from opposite sides.
8. The Definite Integral
The limit of those sums, and the moment it stops being an approximation.
9. The Fundamental Theorem of Calculus
The rate at which area accumulates is the height of the curve — and everything else follows.
10. Properties of Definite Integrals
The symmetry shortcuts, shown as pictures before they are stated as rules.
11. Area Under a Curve
Signed area and total area — two numbers, both correct, answering different questions.
12. Area Between Curves
The axis replaced by a second curve — and the region splitting where they cross.
13. Volumes of Revolution
Rotating a region, and building a volume from discs — which finally derives the cone's one-third.
14. Trapezoidal Rule and Simpson's Rule
What to do when no antiderivative exists — which is most of the time.
15. Improper Integrals
An infinitely long region with a finite area — and a nearly identical one without.