Find its whole area of the cardioid r=a(1-cas(theta))
Answers
Answered by
1
Step-by-step explanation:
3
π
2
areal units.
Explanation:
If the pole r = 0 is not outside the region, the area is given by
(
1
2
)
∫
r
2
d
θ
, with appropriate limits.
The given curve is a closed curve called cardioid.
It passes through the pole r = 0 and is symmetrical about the initial
line
θ
=
0
.
As
r
=
f
(
cos
θ
)
, r is periodic with period
2
π
.
And so the area enclosed by the cardioid is
(
1
2
)
∫
r
2
d
θ
, over
θ
∈
[
0
,
2
π
]
.
(
1
2
)
(
2
)
∫
(
1
+
cos
θ
)
2
d
θ
,
θ
∈
[
0
,
π
]
, using symmetry about
θ
=
0
=
∫
(
1
+
cos
θ
)
2
d
θ
,
θ
∈
[
0
,
π
]
=
∫
(
1
+
2
cos
θ
+
cos
2
θ
)
d
θ
,
θ
∈
[
0
,
π
]
=
∫
(
1
+
2
cos
θ
+
1
+
cos
2
θ
2
)
d
θ
,
θ
∈
[
0
,
π
]
=
[
3
2
θ
+
2
sin
θ
]
+
(
1
2
)
(
1
2
)
sin
2
θ
]
,
between limits
0
and
π
=
3
π
2
+
0
+
0
Similar questions
Social Sciences,
3 months ago
Math,
3 months ago
English,
6 months ago
Science,
10 months ago
Math,
10 months ago