Math, asked by vinayprasad33, 5 months ago

Find the area bounded by the cardioid r=a(1+cos tetha)​

Answers

Answered by Anonymous
1

Step-by-step explanation:

The cardioid r=a(1+cosθ) is ABCOB

A and the cardioid r=a(1−cosθ) is OC

BA

B

O

Both the cardioids are symmetrical about the initial line OX and intersect at B and B

∴ Required Area=2Area OC

BCO

=2[areaOC

BO+areaOBCO]

=2[(∫

0

2

π

2

1

r

2

dθ)

r=a(1−cosθ)

+∫

2

π

π

((1+cosθ)

2

dθ)

r=a(1+cosθ)

]

=a

2

[∫

0

2

π

(1−2cosθ+cos

2

θ)dθ+∫

2

π

π

(1+2cosθ+cos

2

θ)dθ]

=a

2

[∫

0

π

(1+cos

2

θ)dθ−2∫

0

2

π

cosθdθ+2∫

2

π

π

cosθdθ]

=a

2

[∫

0

π

(1+

2

1+cos2θ

)dθ−2∣sinθ∣

0

2

π

+2∣sinθ∣

2

π

π

]

=a

2

[

2

3

θ+

4

sin2θ

0

π

−2(1−0)+2(0−1)]

=(

2

−4)a

2

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