Practice question · Put in order
Order the steps for evaluating over a triangle by iterated integration.
- Integrate the inner variable, treating the outer one as constant
- Substitute the inner limits, leaving a function of the outer variable
- Integrate the outer variable over its fixed range
- Choose an order, e.g. inside from 0 to , then outside from 0 to 1
- Sketch the region and read off its boundary curves
Hints
- The sketch is not optional, inner limits usually depend on the outer variable, and only the picture reveals how.
- Work from the inside out, like nested parentheses.
Show the answer
- Sketch the region and read off its boundary curves
- Choose an order, e.g. inside from 0 to , then outside from 0 to 1
- Integrate the inner variable, treating the outer one as constant
- Substitute the inner limits, leaving a function of the outer variable
- Integrate the outer variable over its fixed range
Why
Iterated integration slices a 2D problem into 1D strips: the inner integral sums one strip; the outer integral sums the strips. Getting the region’s limits right (from the sketch!) is where every double-integral battle is won or lost.
Practise Double Integrals
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