Numerical Integration

Calculus 3 · Axiom Academy

LESSON Numerical Integration in Multiple Dimensions Approximating a double integral as a grid of prism volumes — then why grids fail in high dimensions and randomness wins. 1. Sampling on a Grid: Boxes Under a Surface A double integral is the volume under the surface z = f(x,y) . To approximate it, lay a grid over the region R , sample f , and add up the volumes of the little boxes (prisms): each box's base is one grid cell and its height is the sampled value. Refine the grid and the box-sum closes in on the true volume. To approximate over a rectangle : Divide [a,b] into n subintervals (even n ) Divide [c,d] into m subintervals (even m ) Apply Simpson's rule in the x -direction at each y -value Combine those results with Simpson's rule in y Every scheme is this same picture — a grid of samples weighted and summed. Simpson's uses the coefficients in each direction; the midpoint rule below just weights every box equally. grid quadrature — a weighted sum of samples the test surface's exact volume (the target) 2. The Curse of Dimensionality Extending the grid to 2D works well, but each new dimension is brutal: a grid with n points per axis needs n^ d points in d dimensions. The count grows exponentially . n points along a line — e.g. 100 points points in a grid — e.g. 10 , 000 points points in a cube — e.g. 1 , 000 , 000 points n^ 10 points in a hypercube — e.g. 10^ 20 points

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