The Fundamental Theorem of Calculus and Definite Integrals

Think of the definite integral as the limit of adding up infinitely many thin rectangles under a curve. That sum is hard to compute directly. The Fundamental Theorem gives a shortcut: find any function F whose derivative is f, then compute F(b) - F(a). This works because F's derivative tells you the instantaneous rate at which area accumulates as x moves right. If F tracks accumulated area up to x, then the area between a and b is just how much F grew over that interval, which is F(b) - F(a). Antidifferentiating undoes differentiating, so this one subtraction replaces an infinite sum.

This same idea is called the net change theorem when f is a rate instead of a height. If f(t) is velocity, the integral of f from a to b gives displacement: total position change, not total distance. If f(t) is a rate of water flowing into a tank in gallons per minute, the integral gives the net gallons added between two times. The theorem says total change equals the integral of the rate of change, which is why units of f times units of x always match the units of the answer.

This topic is the payoff for everything before it in the unit: every antiderivative rule, every technique for finding F, exists so you can plug into F(b) - F(a) and get a number. Once you have the Fundamental Theorem, the rest of BC integration is about building a bigger toolkit of antiderivatives, including integration by parts and partial fractions, so you can apply this same two-step process to harder integrands.