7.4

Lengths of Curves

Quick Review

2

2

2

2

3

1. Simplify the function.

a. 4 4 on 1,5

b. 1 cot on 1,2

1c. 1 on 4,12

4

2. Identify all values of for which the function fails to

be differentiable.

a. ( ) -1

b. ( ) 3

c. ( )

x x

x

x

x

x

f x x

f x x

f x

21 2x x

x2

xcsc

x

x

2

42

1

0

1

What you’ll learn about A Sine Wave Length of a Smooth Curve Vertical Tangents, Corners, and Cusps

Essential QuestionHow can we use definite integrals to find the

length of a smooth curve?

Example The Length of a Sine Wave1. What is the length of the curve y = sin x from x = 0 to x = 2?

Partition [0, 2] into intervals so short that the pieces of curve lying directly above the intervals are nearly straight.Each arc is nearly the same as the line segment joining its two ends.The length of the segment is:

2 2 kk yx

The sum over the entire partition approximates the length of the curve.

2 2 kk yx

Rewrite as a Riemann’s sum.

k

kkk x

xyx

2 2 k

k

kk xx

yx

2

2 2

kk

k xx

y

2

1

Example The Length of a Sine Wave

Rewrite the last square root as a function evaluated at some c in the kth subinterval.

Use the Mean-Value Theorem for differentiable function to obtain the sum:

kk xc 2nsi1

Take the limit as the norms of the subdivisions go to zero:

dxxL 2

0

2nsi1

dxx 2

0

2cos1

NINT ,cos1 2x ,x ,0 2

64.7 3units

Arc Length: Length of a Smooth Curve

b

adx

dx

dyL

2

1

If a smooth curve begins at ( a, c ) and ends at ( b, d ), a < b, c < d, then the length (arc length) of the curve is:

if y is a smooth function of x on [ a, b];

d

cdy

dy

dxL

2

1 if x is a smooth function of y on [ c, d].

Example Applying the Definition

2. Find the length of the curve y = x 2 for 0 < x < 1.

dx

dyx2 This is continuous on [0, 1].

L 1

0 221 x dx

NINT ,21 2x ,x ,0 1

479.1

Example A Vertical Tangent

3. Find the length of the curve between (–1, –1) and ( 1, 1). 3

1

xy

Because the derivative is undefined at x = 0, change the equation to x as a function of y.

3yx dy

dx23y

L 1

1 2231 y dy

NINT ,3122x ,x ,1 1

096.3

Example Getting Around a Corner4. Find the length of the curve y = | x + 1| for –2 < x < 1.

Because the derivative is undefined at x = –1, change the equation to a piecewise function.

1 xy

,1x 1 if x

,1 x 1 if x 1dx

dy

1dx

dy

L

1

2 211 dx

1

1 211 dx

1

2 2

x 1

1 2 x

2 22 2 2 23

Pg. 416, 7.4 #1-29 odd and

#35, 37