Integration and the fundamental theorem

What you’ll learn

The first lesson found a circle’s area by slicing it into rings and adding them up, integration before we had the word. The last eight lessons built differentiation; this one formalizes the other half, accumulation, and states the theorem that binds them. The single capability it builds: compute an integral as an antiderivative, and state the fundamental theorem of calculus.

You will define the definite integral ∫_a^b f(x) dx as the area under the curve, made precise as the limit of Riemann sums (thin rectangles f(x_i)·Δx as the width shrinks to zero, the slice-and-add of lesson 1 written carefully). Then comes the fundamental theorem: if F is an antiderivative of f (F' = f), then ∫_a^b f(x) dx = F(b) - F(a). To accumulate, you do not sum rectangles; you find a function whose rate of change is f and subtract its endpoint values, so differentiation and integration are inverse operations. You will see that antiderivatives are your derivative rules run backward (∫ x^n dx = x^(n+1)/(n+1) + C, with 1/x the n = -1 exception giving ln|x| + C), work several integrals (∫_0^1 x² = 1/3, ∫_0^π sin x = 2, and ∫_0^R 2πr dr = πR² to close the opening circle), and distinguish definite integrals (a number) from indefinite ones (a function F(x) + C).

Where this fits

This is lesson 10 of Phase 3 (Integration and approximation) and its opener. It is the formal answer to lesson 1’s circle-area derivation, and it rests on the limit foundation made precise in lesson 9 (the integral is a limit of Riemann sums). Its antiderivatives reuse every derivative rule from Phases 1 and 2 in reverse. The next lesson (11) unpacks geometrically why the fundamental theorem is true, and the track then turns to higher-order derivatives and Taylor series. Integration is also the calculus prerequisite for continuous probability across the AI tracks.

Before you start

Prerequisite (within this track): lesson 9, Limits, done carefully, since the integral is defined as a limit of Riemann sums. You also want every derivative rule from the track fresh (power, trig, e, and the d/dx(ln x) = 1/x result from lesson 8), because antiderivatives are those rules reversed. Comfort evaluating a function at two endpoints and subtracting is the only new skill; no coding, nothing installed. The practice is pen and paper.

By the end, you’ll be able to

• Define the definite integral as the limit of Riemann sums (the area under a curve)
• State and apply the fundamental theorem of calculus, integral from a to b of f = F(b) - F(a)
• Compute integrals by finding antiderivatives (derivative rules run backward), including the n = -1 case that gives ln|x|
• Distinguish definite from indefinite integrals and explain why the indefinite integral carries a + C

Time and difficulty

• Read time: about 12 minutes
• Practice time: about 13 minutes (computing definite integrals via antiderivatives, an indefinite-integral drill, and flashcards)
• Difficulty: standard