Multiple Integration Summary

Calculus 3 · Axiom Academy

How double and triple integrals extend integration to higher dimensions — measuring volumes, masses, and averages across regions in space. Dimension extension: multiple integrals carry the single-variable idea systematically into 2D and 3D — accumulating a quantity over a region rather than an interval. Coordinate choice matters: the right coordinate system can turn an impossible integral into a simple one — match the system to the region's symmetry. Order is flexible, bounds are not: Fubini's theorem lets you swap integration order, but the limits must be re-derived for the new order. Every change of variables carries a Jacobian: , — never drop the volume-element factor. Applications give meaning: mass, center of mass, and moments turn an integral into a physical, geometric quantity. Sketch first: drawing the region before writing bounds prevents most setup errors. Extends single integration to two dimensions: the signed volume under the surface z = f(x,y) over a region R . Fubini's theorem lets you evaluate it as an iterated integral in either order when f is continuous. Watch out for: on a general region the bounds become functions, not constants. Core Concept General Regions (Type I / II) A Type I region is bounded by and (integrate dy then dx ). A Type II region uses and (integrate dx then dy ). When to use: pick the slicing whose inner bounds are simplest. Watch out for: a complex region may need splitting into simpler pieces. Core Concept Polar Coordinates

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