Chapter 5: Applications of Integration
Section 5.1: Area of a Plane Region
Essentials
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
Bounds on Region
Area
fx≥0 on a,b
y=fx
y=0
x=a
x=b
∫abfx ⅆx
fx≥0 on a,c
fx≤0 on c,b
∫acfx ⅆx−∫cbfx ⅆx
fx≥gx on a,b
y=gx
∫abfx−gx ⅆx
fx≥gx on a,c
gx≥fx on c,b
∫acfx−gx ⅆx+∫cbgx−fx ⅆx
Table 5.1.1 Computing area of plane regions
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 x, and its width given by an increment in x.
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 xy, and its "thickness" determined by an increment in y. See the Examples below for illustrations of this approach to calculating the areas of plane regions.
Examples
Example 5.1.1
Calculate the plane area bounded by the graph of fx=6+x−x2 and the x-axis.
Example 5.1.2
Calculate the plane area bounded by the graph of fx=x3−7 x2+5 x+4 and the x-axis.
Example 5.1.3
Calculate the area bounded by the graphs of fx=x and gx=x2.
Example 5.1.4
Calculate the area of the region R bounded by the graphs of fx=1,gx=1−3 x/2, and hx=x.
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