Section 8.1 MATH 1152Q Arc Length Notes
Arc Length
Suppose that a curve C is defined by the equation yfx= ( ), where f is continuous and axb££. We obtain a polygonal approximation to C by dividing the interval [ab, ] into n subintervals with endpoints x0 , x1 , , xn and equal width Dx . If yfxii= ( ), then the point Pi
(xyii, ) lies on C and the polygon with vertices P0 , P1 , , Pn is an approximation to C .
The length L of C is approximately the length of this polygon and the approximation gets better as we let n increase. Therefore, we define the length L of the curve C with equation yfx= ( ), axb££, as the limit of the lengths of these inscribed polygons (provided the limit exists).
n
L = lim PPii-1 n®¥ å i=1
The definition of arc length given above is not very convenient for computational purposes, but we can derive an integral formula for L in the case where f has a continuous derivative. Such a function f is called smooth because a small change in x produces a small change in fx'( ).
If we let Dyyyiii=--1, then 22 PPii-1 =(xxii---11) +( yy ii-)
22 =(Dxy) +(Di ) éù2 2 æöDyi =(Dx) êú1+ç÷ ëûêúèøDx 2 æöDyi =+1 ç÷ ×Dx èøDx
By applying the Mean Value Theorem to f on the interval [xxii-1, ], we find that there is a * number xi between xi-1 and xi such that * fx( ii) - fx( --11) = f'( x iii)( x- x)
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Thus we have 2 æöDyi PPii-1 =+1 ç÷ ×Dx èøDx 2 =+1'éùfx* ×D x ëû( i )
Therefore, n
L = lim PPii-1 n®¥ å i=1 n 2 * =+lim 1éùfx ' i ×D x n®¥ å ëû( ) i=1 b 2 =+1'éùfx( ) dx òa ëû
2 The integral exists because the integrand 1'+ ëûéùfx( ) is continuous. Thus we have the following theorem.
Formula The Arc Length formula
If f ' is continuous on [ab, ], then the length of the curve yfx= ( ), axb££, is
L = ______.
Similarly, if a curve has the equation xgy= ( ), cyd££, and gy'( ) is continuous, then
L = ______.
Example: Find the length of the curve yx23= for 14££x .
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The Arc Length Function
If a smooth curve C has the equation yfx= ( ), axb££, let sx( ) be the distance along C from the initial point P0 (afa, ( )) to the point Q (xf, ( x)) . Then s is a function, called the arc length function. x 2 sx( ) =+1'éùft( ) dt òa ëû
Special Problem
Example: 1 éù1 Find the length of the curve fx( ) =+ x3 on the interval ,2 . 12x ëûêú2
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