Numerical Integration

When a function has no clean antiderivative, computers step in with numerical integration. The simplest method, the trapezoidal rule, approximates the area under a curve using trapezoids instead of rectangles, cutting error sharply for little extra work.

More advanced methods like Simpson's rule fit parabolas through groups of three points, reaching near-exact answers with surprisingly few slices. Weather models, rocket trajectories, and financial pricing engines all rely on these techniques.

The core idea never changes: slice, approximate, sum, and shrink the slices. Numerical integration is Archimedes' polygon method reborn as software.