How do you find the region inside cardioid r=1+cos(θ) and outside the circle r=3cos(θ)?
Answers
Answered by
2
Find the intersection points of the curves hence we have that
3
cos
θ
=
1
+
cos
θ
⇒
cos
θ
=
1
2
⇒
θ
=
±
π
3
The saded area is
(cardiod area from pi/3 to pi)-(cricle area from pi/3 to pi/2)
The cardiod area is
∫
π
π
3
1
2
⋅
(
1
+
cos
θ
)
2
d
θ
=
π
2
−
9
6
⋅
√
3
and the circle area is
∫
π
2
π
3
1
2
⋅
(
3
⋅
cos
θ
)
2
d
θ
=
(
3
π
8
)
−
9
16
⋅
√
3
Hence the shaded area is
π
8
The total amount is
2
π
8
=
π
4
A graph for the curves is
Attachments:
Similar questions