Find the perimeter of the loop of the curve 3ay2
= x(x - a)
Answers
Answer:
Step-by-step explanation:
The curve is symmetrical about the x−axis and the loop lies between the limits x=0 and x=a
We have y=
3a
x(x−a)
∴
dx
dy
=
3a
1
[
2
3
x
2
1
−
2
a
x
2
−1
]
=
2
3a
1
x
3x−a
∴ Perimeter of the loop=2∫
0
a
1+(
dx
dy
)
2
dx (By symmetry)
=2∫
0
a
[1+
12ax
(3x−a)
2
dx]
=2∫
0
a
12ax
9x
2
+6ax+a
2
dx
=
3a
1
∫
0
a
x
3x+a
dx
=
3a
1
∫
0
a
(3x
2
1
+ax
2
−1
)dx
=
3a
1
∣
∣
∣
∣
∣
∣
2
3
3x
2
3
+a
2
1
x
2
1
∣
∣
∣
∣
∣
∣
0
a
=
3a
1
(4a
2
3
)
=
3
4a