Mr. Integral

THE BIG IDEA

Double Integrals

How do we accumulate over an area?

Extend weighted sums from intervals to planar regions. Draw the region before choosing bounds, and change coordinates when a simpler description outweighs the extra area-scaling factor.

Begin the first lesson →See the lesson route ↓

4 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Area accumulation
  2. computing it requires02Region bounds
  3. radial symmetry suggests03Polar coordinates
  4. local area scaling generalizes to04General coordinate changes

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Lessons in this unit4 lessons
  1. 01Double Integrals and AccumulationInterpret a double integral as a limit of area-weighted samples and evaluate it by iteration.
  2. 02Drawing Regions and Changing Integration OrderTranslate geometric boundaries into bounds before reversing an iterated integral.
  3. 03Polar Coordinates in Double IntegralsConvert a radial region, integrand and area element together.
  4. 04General Changes of Variables and JacobiansMeasure local area scaling and map a simple parameter region to a harder physical region.