Processing math: 100%
SOLUTION 13: Compute the area of the
region enclosed by the graphs of the equations y=x3 and y=4x . Begin by finding the points of
intersection of the two graphs. From y=x3 and y=4x we get that
x3=4x ⟶
x3−4x=0 ⟶
x(x2−4)=0 ⟶
x(x−2)(x+2)=0 ⟶ x=0,x=2, or x=−2
Now see the given graph of the enclosed region.
Using vertical cross-sections to describe this region, which is made up of two smaller regions, we get that
−2≤x≤0 and 4x≤y≤x3
in addition to
0≤x≤2 and x3≤y≤4x ,
so that the area of this region is
AREA=∫0−2(Top − Bottom) dx+∫20(Top − Bottom) dx
=∫0−2(x3−4x) dx+∫20(4x−x3) dx
=(x44−4⋅x22)|0−2+(4⋅x22−x44)|20
=( ((0)44−4⋅(0)22)−((−2)44−4⋅(−2)22) )+( (4⋅(2)22−(2)44)−(4⋅(0)22−(0)44) )
=( (0−0)−(4−8) )+( (8−4)−(0−0) )
=4+4
=8
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