Irregular Territories

Calculus 3 · Axiom Academy

A farm bends around a river, so its map is no rectangle. One double integral still tells you the total crop value of the whole irregular plot. A river carves the south edge of a farm plot : it runs 4 km east, its northern ridge road is straight, but its southern boundary is the curved river bank . The soil is richest by the water. Find the plot's total crop value — three moves, one relationship: total = ∬ density over the region . Slide the strip east and west. A rectangle would have a flat bottom — but this plot's bottom is the river , a curve, so each vertical strip runs from the river up to the road, and its height changes with x . That is exactly what means. Add up the value, strip by strip Now colour the plot by crop value — deep by the river, pale by the road, from the density (thousands of dollars per km²). Slice the region into n strips and add up value = density area, one strip at a time. As n grows, the running total closes in on the double integral — the plot's true total value. One number for the whole territory Here is why the integral is worth the trouble: it collapses the whole irregular plot into one number you can act on . Drag how sharply value drops as you leave the river — the falloff m in — and the plot's total value re-totals instantly across every curved strip at once.

This is the written version of the interactive lesson above. See the full Calculus 3 course.