The length of upper half of the cardioide r = a(1 + cos 0) where 8 varies from 0 to rr using r² + (a) = 2a² (1 + cos8) is
Answers
Answer:
The cardioid is symmetrical about the initial line and for its upper half,θ increases from 0 to π
Also,
dθ
dr
=−asinθ
∴Length of the curve=2∫
0
π
[r
2
+(
dθ
dr
)
2
]
dθ
=2∫
0
π
[a(1+cosθ)]
2
+(−asinθ)
2
dθ
=a∫
0
π
1+cos
2
θ+2cosθ+sin
2
θ
dθ
We know that sin
2
θ+cos
2
θ=1
⇒2a∫
0
π
2(1+cosθ)
dθ
We know that 1+cosθ=2cos
2
2
θ
=2a∫
0
π
2×2cos
2
2
θ
dθ
=4a∫
0
π
cos
2
θ
dθ
=4a
∣
∣
∣
∣
∣
∣
∣
∣
2
1
sin
2
θ
∣
∣
∣
∣
∣
∣
∣
∣
0
π
=8a(sin
2
π
−sin0)
=8a
∴ Length of upper half of the curve is 4a
Also length of the arc AP from 0 to
3
π
=∫
0
3
π
2(1+cosθ)
dθ
=2a∫
0
3
π
cos
2
θ
dθ
=4a
∣
∣
∣
sin
2
θ
∣
∣
∣
0
3
π
=4a(sin
6
π
−0)
=4a×
2
1
=2a=half the length of upper half of the cardioid.
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