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MATH
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Methods from Regions and order of integration

All methods from Regions and order of integration.

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Set up an integral over a planar region

To set up a double integral over Ω\Omega:

  1. sketch the region and every boundary curve;
  2. decide whether vertical or horizontal slices give simpler bounds;
  3. find the range of the outside variable;
  4. fix that variable and find the two inner bounds;
  5. write the inner integral first, then integrate from the inside outward;
  6. split the region if one pair of inner bounds does not describe every slice.

Changing the order describes the same region by different slices, so the new limits must be read from the boundary, not by swapping the old ones.

  1. [1]:Vertical-slice regions
  2. [2]:Horizontal-slice regions
  3. [3]:A triangular region
  4. [4]:Reverse the order to find an antiderivative
  5. [5]:A region that must be split

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Set up an integral over a planar region