This page assumes nothing. Before you read the proofs in the parent topic, every squiggle, letter, and idea it leans on is built here from the ground up, in an order where each piece rests on the one before it.
Picture it. Pick an x, walk up to the curve, read the height — that height isf(x).
Why the topic needs it. The FTC is a statement about one continuous curve f: it relates the area beneath that curve to the height of that curve. Both ideas live on the same picture.
[a,b]
means "all numbers from a to b, including a and b" — a closed interval, the horizontal stretch we work over.
How do we measure a curvy area? We approximate it with thin rectangles, then let the rectangles get infinitely thin. That summing-up process is written with a special stretched-S symbol.
Picture it. A tall column of near-zero width, area f(x)dx, sliding from a to b; add them all up and you get the total shaded area.
Why the topic needs it. The left side of FTC Part 2 is this symbol. To prove anything about it we must know it means "limit of rectangle sums" — that is exactly what Riemann Sums and the Definite Integral builds in full.
∫aafdt
equals 0 because the region from a to a has zero width.
Picture it.t is a finger sliding along the base from a rightward; x is where you stop the finger. Change where you stop, and you change the shaded area — so the area is a function of x, which is why we name it F(x).
Why the topic needs it.F(x) — the "area-so-far" function — is the star of Part 1. Getting the two letters straight is the difference between a valid proof and nonsense.
F(x)=∫axf(t)dt in words
"the area under f from the fixed left edge a up to the moving right edge x."
Picture it. On any closed stretch of an unbroken curve there is a genuine peak and a genuine valley — the curve can't "run off to a value it never quite hits."
Why the topic needs it. The thin sliver of area from x to x+h is squashed between the shortest possible rectangle mh⋅h and the tallest Mh⋅h. EVT guarantees those two heights exist. See Extreme Value Theorem.
Picture it. A sandwich: press the top slice and bottom slice together and whatever is between them has no choice but to meet at the same spot.
Why the topic needs it. As h→0, both mh and Mh slide toward f(x); the Squeeze Theorem forces the difference quotient (the thing between them) to equal f(x) — that is Part 1. Detailed in Squeeze Theorem.
Picture it.hF(x+h)−F(x) is the slope of the line joining two nearby points on F's graph; as h→0 that line becomes the tangent, and its slope is F′(x).
Why the topic needs it. Part 1 claims F′(x)=f(x). To prove it we must start from the definition of F′ — this limit — and nothing else. The symbol limh→0 just means "the value being approached as h shrinks to 0."
hF(x+h)−F(x) in words
"how much the area grew, divided by how much wider the region got" — the average height of the newly added strip.
Why the topic needs it. Part 2 says: to get ∫abf, find any antiderivative G and compute G(b)−G(a). Reversing the power/rules to find G is the whole content of Antiderivatives and Indefinite Integrals.
Picture it. Draw the straight chord from the start point to the end point; somewhere in between, the curve runs parallel to that chord.
Why the topic needs it. Part 2 needs "if (F−G)′=0 everywhere then F−G is constant." That is not obvious — MVT proves it: with G′(c)=0, the formula gives G(b)−G(a)=0 for any endpoints, so G never changes. Full treatment in Mean Value Theorem.
Why the topic needs it. When the upper limit is not plain x but something like x2 or x3, e.g. dxd∫0x2f, Part 1 handles the "H′" piece and the Chain Rule supplies the extra factor dxd(x2)=2x. Miss it and you get the wrong answer.
