Calculus Volume 3 · free preview
Double Integrals over General Regions
General plane regions have variable limits in the iterated integrals, the type one and type two descriptions integrating in either order with the boundary curves as limits, and reversing the order re-describing the region.
1. Type one and type two regions
Type one and type two regions
2. Reversing the order
Reversing the order
3. Splitting and the area
Splitting and the area
4. Probability interpretation
Probability interpretation
5. Type one limits
A type one region's iterated integral has which limits?
6. Reversal purpose
Reversing the order of integration is most valuable when it does what?
7. Region splitting
A region neither type one nor type two requires which treatment?
8. Area as integral
The area of a plane region D equals which double integral?
9. Choosing the type
A region bounded by a curve and a vertical line, with the curve single-valued in y for each x, is naturally which type?
10. Reversal log limit
Reversing the region under y = e^(-x) gives an x upper limit solving y = e^(-x) for x, which is what? Answer like -ln y.
11. Region between curves area
The region between y = x^2 and y = 2x has area the integral of 2x minus x squared from 0 to 2. As a number, what is the area? Answer with a number.
12. Type two inner variable
In a type two integral, the inner integration is over which variable? Answer with one lowercase letter.
13. Sketching necessity
Why is sketching the region considered indispensable before setting up a general double integral?
14. Volume between surfaces
The volume between two surfaces over D is the double integral of the difference. As two words, what difference? Answer like top minus.
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