Find the area between the cardioids r = 2a(1−cos θ) and r = 2a(1+cos θ).
Suppose the area lying inside the cardioids is revolved about the initial
line θ = 0. Find the volume of the solid thus generated.
Answers
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
solution
Step-by-step explanation: