Integrating Over Two Dimensions

A double integral extends the concept of area under a curve to volumes under a surface. Given a function $f\left(x,,,y\right)$, the double integral over a rectangular region $[a,b]$ × $[c,d]$ computes the total volume between the surface and the plane. Analytically evaluating these integrals can be tricky, making numerical approaches invaluable.

Numerical Scheme

This calculator partitions the rectangle into a grid and applies the midpoint rule in both directions. Each small rectangle contributes $f\left(x\_i,,,y\_j\right)$ times the area of the subrectangle. Summing all contributions yields an approximation to the true integral. You can refine the grid by adjusting the number of subdivisions in the code to trade speed for accuracy.

Applications

Double integrals appear throughout physics and engineering. They compute mass of a plate with varying density, heat over a region, or probabilities for bivariate distributions. Being comfortable with numerical approximations helps tackle problems without simple antiderivatives.