8.5 Arc Length ( Rectification )
The method of finding the length of the arc of a curve is called the rectification.
For arc length, the function and its derivative must both be continuous on the closed interval.
If y = f (x) and f’ (x) are continuous on [ a, b ], then the arc length ( L ) of f (x) on [ a,b ] is given by
Similarly for, x = f (y) and f’ (y) are continuous on [ a,b ], then the arc length (L) of f (y) on [ a,b ] is given by
Example 29 Find the length of the arc of f (x) = x^{3/2} on [ 0, 5 ].
Solution : both are continuous on [ 0, 5 ].
then length of arc of
Example 30 Find the length of the arc of the parabola y^{2} = 12x cut off by the Latus rectum.
Solution : y^{2} = 12 x ®
Parabola.
Comparing with y^{2} = 4 ax, we see that 4 a = 12 \
a = 3 \
AS = SL = 6 \ coordinates of L are (3,6)
Example 31 Find the length of the curve y^{2} = (2 x  1 )^{3}, cut off by the line x = 4.
Solution : y^{2} = (2 x  1 )^{3} is a curve (i) symmetrical about the xaxis (ii) not passing through (0,0) (iii) it cuts the xaxis, where y = 0 \
x = ˝ \
It's vertex is at ( ˝ , 0) (iv) It does not cut the yaxis as, taking x = 0 we get y = ± i (v) No asymptotes. (vi) (2x  1)^{3} positive \
y^{2} ³
0 \
x ³
˝ \
The curve lies in only 1st and 4th quadrants
Now
[next page]
