The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus simply says that finding areas and finding slopes are opposite operations, so each one undoes the other. All we're really saying is: if you know how fast something is changing, the total change is the difference between where it ended and where it started. The first formula has three pieces: $\int_a^b f(x)\,dx$ (the area, the hard thing we want), $F$ (any antiderivative of $f$, from the last few lessons) and $F(b) - F(a)$ (its value at the end minus its value at the start). The second formula says the same thing from the other side. Two lines, but they connect the whole course.

Think about a car trip. Your speedometer tells you $v(t)$, and the distance you've driven is the area under $v$. But your odometer also tells you the distance, and its rate of change is your speed, so the odometer reading is an antiderivative of $v$. To find how far you went between two times, you don't add up rectangles: you just subtract the two odometer readings. That's $F(b) - F(a)$.

Why it's true. Let $A(x) = \int_a^x f(t)\,dt$ be the area under $f$ from $a$ to $x$. Nudge $x$ by $h$: the area grows by a thin strip of height about $f(x)$ and width $h$. So $A'(x) = f(x)$, and $A$ is an antiderivative of $f$. Any other antiderivative differs by a constant, which cancels in $F(b) - F(a)$.

Notation. $F(b) - F(a)$ is written $\Big[F(x)\Big]_a^b$. You can drop the $+C$; it cancels.

With the chain rule. If the upper limit is a function, $\frac{d}{dx}\int_a^{g(x)}f(t)\,dt = f\big(g(x)\big)\,g'(x)$.

This is the theorem that makes calculus work: a limit of sums of infinitely many thin rectangles reduces to two evaluations of an antiderivative.

Every time an engineer computes total energy from power, total water from a flow rate, or distance from a speed log, they're using the Fundamental Theorem: integrate the rate, get the total change.