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Integrals from the Ground Up: Meaning, Method, and Power
by Gentle-Moon-1001A Socratic journey from 'what is an integral, really?' to computing them fluently and using them on real problems. You start with the area problem and the idea of accumulation, discover why the Fundamental Theorem of Calculus makes derivatives and integrals inverse operations, then master the core techniques — substitution, integration by parts, partial fractions — before applying integrals to areas, volumes, and even infinite regions and probability.
Library
Part 1: What Integrals Mean
Lesson 1
The Two Rules Nobody Taught You (And Why Your Answers Are Wrong)
Your derivative toolkit handles x^5 and sin(x) just fine — so why does it silently give the wrong slope the moment two functions get tangled together?
StartLesson 2
The Shape That Broke Every Area Formula
A region with one curved edge defeats every area formula you've ever learned — so how did mathematicians pin down its area exactly, to the last digit?
StartLesson 3
Integrals as Accumulation
Integrals aren't just about area — they add up anything that accumulates: distance, water, money, energy.
Locked till you take the prior lessonLesson 4
The Fundamental Theorem of Calculus
The most surprising result in calculus: areas and derivatives are secretly inverse operations. Discover why.
Locked till you take the prior lessonLesson 5
Antiderivatives & Basic Integrals
Build your integral toolkit by running every derivative you know in reverse — and meet the mysterious +C.
Locked till you take the prior lessonPart 2: Techniques
Lesson 6
u-Substitution
The chain rule in reverse — the workhorse technique that unlocks integrals your basic table can't touch.
Locked till you take the prior lessonLesson 7
Integration by Parts
What do you do with ∫x·eˣ dx? Run the product rule backwards and trade a hard integral for an easier one.
Locked till you take the prior lessonLesson 8
Partial Fractions
How do you integrate 1/(x²−1)? Break the fraction into simpler pieces you already know how to handle.
Locked till you take the prior lessonLesson 9
Knowing Which Tool to Use
Three techniques, one integral in front of you — which do you reach for? Build the pattern-recognition experts use.
Locked till you take the prior lessonPart 3: Applications
Lesson 10
Areas Between Curves
Two curves, one enclosed region — learn to turn a picture into an integral and get the exact area.
Locked till you take the prior lessonLesson 11
Volumes of Revolution
Spin a curve around an axis and you get a 3D solid — slice it into disks and integrate your way to its exact volume.
Locked till you take the prior lessonLesson 12
Improper Integrals & a Taste of Probability
Can an infinitely long region have a finite area? Sometimes yes — and that paradox is why the bell curve works.
Locked till you take the prior lesson