
Applications of Integration 15-31
Example 15.24 Find the length of the arc of parabola y
2
= 4ax extending from the vertex (0, 0) to an
extremity of the lotus rectum (a, 2a).
Solution: Here we shall apply the formula involving
dx
dy
in place of
dy
dx
, so that
Given y
2
= 4 ax
x
y
a
=
2
4
dx
dy
y
a
=
2
S
dx
dy
dy
y
a
dy
a
a
= +
∫
= +
∫
=
1
1
4
1
2
0
2
2
2
0
2
( )
22
2
0
2
2 2
2
2 2
2
1
2
4
4
2
4
+
∫
= + + + +
{ }
y
a
a
y a
a
y y a
a
log
+
∫
= +
{ }
+ −
0
2
2 2
2 1 2 1
a
a x dx
a
using
log .
Example 15.25 Find the length of the arc of y = ln (sin x) between
p p
£ £x .
Solution: Given curve is y = ln (sin x) where
p p
£ £x .
Thus,
dy
dx
x
x
x= =
cos
sin
cot
Therefore, the ...