Table 5.1.1 lists plane regions bounded by exactly two plane curves, and the definite integrals that will calculate the areas of those regions.
Conditions
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Bounds on Region
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Area
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on
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on
on
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on
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on
on
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Table 5.1.1 Computing area of plane regions
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If the plane region is bounded by more than two curves, it can be subdivided into regions bounded by just two curves, and the listings in Table 5.1.1 can be applied to the subregions.
The area of each region in Table 5.1.1 is found with a definite integral that uses "vertical strips." The elementary rectangle in the Riemann sum underlying each such definite integral has its height determined by the value of a function of , and its width given by an increment in .
For some regions, it is possible to obtain the area with a "horizontal strip." In these cases, the elementary rectangle in the Riemann sum has its "height" determined by the value of , and its "thickness" determined by an increment in . See the Examples below for illustrations of this approach to calculating the areas of plane regions.