Question

12. Prove that the area common to the circle r =a and the cardioid r = a (1 + cos thetha) is a square (5pie upon 4-2)




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Answer 1

$\huge\star\underline\mathfrak\red{Answer:-}$

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

3π

−4)a

2