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4.3 AREA
ofPOLAR
REGIONS
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Area of a sector
Area of a sectorof a circle:
where is in
radians
θ
2
2
1
r A=
θ
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Area of a polar region
Suppose for 0 θf r
]
θd f A 2 2
1
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Suppose
for
0 θgf
]
]
θ
d gf A 2 2 2
1
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Example. Setup the definite inte!ral
that sol"es for the area of the shaded
re!ion#
θcosr $ $
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Solution:
]
∫
θd f A 2 2
1
[ ]
∫
θd cos 2 $ $ 2
1
0
π2
2
%$ π
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Example. Setup the definite inte!ral
that sol"es for the area of the shaded
re!ion#
θ2 cosr
&
π
θ
1 A
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Solution:
]
∫
θd f A 2
1 2
1
[ ]
∫
θd cos 2 2 2
1
0
&
π
1
'
A A=
=
1(
' π
2
π
=
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Example. Setup the definite inte!ral
that sol"es for the area of the shaded
re!ion#
2 r
2
π
θ
cosr 2 2
2 A 1 A
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Solution:
2 12
A A A
[ ]
∫
θd cos A
2
1 2 2
2
1
[ ]
∫
θd A 2 2 2 2 1
2
π
2
π
0
π
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Example. Setup the definite inte!ral
that sol"es for the area of the shaded
re!ion#
2 r
1 A
θ
cosr 2 2
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Solution:
∫
θd cos 2 2 2 2 2 2
1
12 A A=
π
2
π
]
]
∫
θd gf A
2 2
1 2
1
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Example. Setup the definite inte!ral
that sol"es for the area of the shaded
re!ion#
1r
θ2 sinr
1 A
&π
θ
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Example. Setup the definite inte!ral
that sol"es for the area of the shaded
re!ion#
θcosr 2 1
)r )
2 πθ
2 A
1 A
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Solution:
∫
θd cos A 2 2 1
2 1) 2 1
2 12 A A A
0
)
2 π
[ ]
∫
θd A 2 2 ) 2
1 π
)
2 π
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END
