Calculus
Mrs. Cartledge
Arc length
Phantom’s Revenge Kennywood Park
Sec 7.4
Objectives
Determine the length of a curve.
Area Under a Curve (revisited)
We use the definite
integral to find the
area under a curve.
The limits of
integration are a
and b. The height
of the rectangle is
represented by f(x),
the width by dx and
the sum by the
definite integral.
Sum of Disks as the Volume
(revisited)
b
V    [ R ( x )] dx
2
a
Finding Arc Length
We want to determine the length of the continuous
function f(x) on the interval [a,b] . Initially we’ll need
to estimate the length of the curve. We’ll do this by
dividing the interval up into n equal subintervals each
of width x and we’ll denote the point on the curve at
each point by Pi. We can approximate the curve by a
series of straight lines connecting the points.
The length of one segment
P5
P4
The length of all segments
n
L   ( xi  yi ) 
2
2
i 1
n

i 1

i 1

i 1
( xi  yi ) * Δxi
2
2
xi
n
n
  1  [ f ( x )]2 dx
a
( xi  yi )
2
xi
b
2
2
* Δxi
yi
(1 
) * Δxi
2
xi
2
Definition of Arc Length
Example Applying the
Definition
Find the length of the curve y  x for 0  x  1.
2
dy
 2 x, this is continous on 0,1. Therefore,
dx
L   1   2 x  using NINT
1
0
L  1.479
2
Example
Find the length of the specified curve.
= .881374
Algorithm to Find Arc Length
1. Determine whether the length is with
respect to x or y, and then find the
endpoints for the interval.
2. Find y’(x) or x’(y).
3. Plug the derivative into the formula:
4. Evaluate.
Sample
Find the length of the specified curve.
[1,2]
Sample
Find the length of the specified curve. You
are given the x values but you need c and d!!
Sample
Find the length of the specified curve.
y = sin(x) [0, ¶ ]
Sample
Find the length of the specified curve.
y = (x2 – 4)2 [0, 4 ]
Example A Vertical Tangent
1
3
Find the length of the curve y  x between (-1,-1) and (1,1).
dy
1

is not defined at x = 0. There is a vertical tangent
dx 3x
2
3
at (0,0). Change to x as a function of y in order to make the
tangent at the origin horizontal and the derivative equal to
zero instead of undefined.
dx
Solve y  x for x. x  y and
 3y
dy
1
3
L
1
1
1  3 y
3
2
2
 dy  3.096 using NINT.
2
Sample
y = x1/5 [-1,4]
Closure
Explain the difference in these two
formulas.
Independent Assignment
Notebook: p 485 # 3 - #11 odd. Check
your answers in the back of the book.
Graded Assignment: HW Sec 7.4 in
Schoology. Enter the first ½ of the
answers in Schoology. Due Tuesday, May
15, before class.