Double Integrals over General Regions
Integrate over a plane region by iterating two single integrals, with the inner limits describing the region. Choosing the order of integration is often the whole problem.
What a learner can do afterwards
- Set up an iterated integral for a region bounded by two curves
- Reverse the order of integration and redraw the region to find the new limits
- Compute a volume under a surface over a bounded region
1 · Read
A general region is trapped between two curves, so the inner limits become functions. If y runs from a lower curve to an upper curve while x runs from a to b, write the iterated integral with y inside. Between y = x squared and y = x on [0, 1], the line is on top, so y runs from x squared to x.
Reversing means describing the same region with the other variable outside. The triangle with x outside 0 to 1 and y from x to 1 is really 0 <= x <= y <= 1. With y outside 0 to 1, x runs from 0 to y. Both sets of limits change, so redraw the region.
Evaluate inside out, substituting limits before moving outward. Integrating 1 over a region returns its area: the 2 by 3 rectangle gives 6, and the integral from x = 0 to 2 of y = 0 to x of 1 gives 2. Volume works the same way: height z = 2 over the unit square gives 2.
Picking the wrong order makes life hard. When horizontal slices cut the region into pieces with different tops, try dxdy instead. Check any setup with a constant: integrating 1 must return the geometric area of the region.
Sketch the region, put functional limits inside with constants outside, and reverse the order whenever the shape fights back.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.