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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) = x3/2 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 y2 = 12x cut off by the Latus rectum.

Solution : y2 = 12 x ® Parabola.

Comparing with y2 = 4 ax, we see that 4 a = 12 \ a = 3 \ AS = SL = 6 \ co-ordinates of L are (3,6)

Example 31 Find the length of the curve y2 = (2 x - 1 )3, cut off by the line x = 4.

Solution : y2 = (2 x - 1 )3 is a curve
(i) symmetrical about the x-axis
(ii) not passing through (0,0)
(iii) it cuts the x-axis, where y = 0 \ x = ˝
\ It's vertex is at ( ˝ , 0)
(iv) It does not cut the y-axis as, taking x = 0
we get y = ± i
(v) No asymptotes.
(vi) (2x - 1)3 positive \ y2 ³ 0
\ x ³ ˝
\ The curve lies in only 1st and 4th quadrants

Now

                      

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Index

8.1 Introduction
8.2 Area
8.3 Volumes
8.4 Mean Value
8.5 Arc Length(Rectification)

Chapter 1

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