Processing math: 100%
SOLUTION 23: Compute the area of the region enclosed by the graphs of the equations y=x,y=3x,
and x=2 . Now see the given graph of the enclosed region.
a.) Using vertical cross-sections to describe this region, we get that
0≤x≤2 and x≤y≤3x ,
so that the area of this region is
AREA=∫20(Top − Bottom) dx
=∫20(3x−x) dx
=∫202x dx
=(x2)|20
=(2)2−(0)2
=4
b.) Using horizontal cross-sections to describe this region, which is made up of two smaller regions, we get that
0≤y≤2 and 13y≤x≤y
in addition to
2≤y≤6 and 13y≤x≤2 ,
so that the area of this region is
AREA=∫20(Right − Left) dx+∫62(Right − Left) dx
=∫20(y−13y) dy+∫62(2−13y) dy
=∫20(23y) dy+∫62(2−13y) dy
=(y23)|20+(2y−y26)|62
=((2)23−(0)23)+((2(6)−(6)26)−(2(2)−(2)26))
=(43)+(6)−(4−23)
=4
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