The Fundamental Theorem of Calculus connects two apparently separate ideas. Part 1: if you integrate a function from a fixed point to x, the derivative of that integral is the original function. Part 2: the definite integral of f from a to b equals any antiderivative F evaluated at b minus F at a.
The shaded area = ∫₀² x² dx. By FTC: find antiderivative F(x)=x³/3, then compute F(2)-F(0)=8/3. No summing of rectangles required.
Before this theorem, computing areas required Riemann sums: dividing the region into thin rectangles, summing them all, and taking the limit. The FTC replaces all of that with one subtraction. Newton understood this by 1666 and Leibniz independently by 1675. Their dispute over priority split European and British mathematics for a generation.
Every integral taught in calculus courses uses Part 2: find an antiderivative, evaluate at the endpoints, subtract. This works because differentiation and integration are exact inverses of each other. It is one of the deepest and most useful results in all of mathematics.
Each blue rectangle approximates the area under x². Adding all rectangles and taking the limit gives the exact area. The FTC replaces this entire process: just evaluate x³/3 at x=2 minus x=0.
Work done by a variable force F(x) over displacement from a to b is W = integral from a to b of F(x) dx = P(b) - P(a), where P is the potential energy function satisfying P' = -F. Velocity integrates to displacement; force integrates to impulse. The FTC is what makes these calculations tractable rather than requiring infinite Riemann sums.