7.4 Length of a Plane Curve

y=f(x) is a smooth curve on [a, b] if f ’ is continuous on [a, b].

Where convenient, (3) can also be expressed as

Example Find the arc length of the curve from (1, 1) to3/2y x

Solution: 1/23

2

dyx

dx

2 22 1/2 2

1 1

3 91 ( ) 1 ( ) 1

2 4

b

a

dyL dx x dx x dx

dx

(2,2 2)

9 91 ,4 4

Let u x du dx

9 131 1

4 4x u 9 22

2 1 24 4

x u

22/422/4 1/2 3/2 3/2 3/2

13/413/4

4 8 8 22 13( ) ( )

9 27 27 4 4

22 22 13 132.09

27

L u du u

Arc Length Formula for Parametric Curves

Example: Find the circumference of a circle of radius 2 from the parametric equations

2cos , 2sin (0 2 )x t y t t

Solution:

2 2

2 2 2

0

2 2

00

( ) ( )

( 2sin ) (2cos )

2 2

4

b

a

dx dyL dt

dt dt

t t dt

dt t

7.5 Area of a Surface of Revolution

y=f(x)

a b

Where convenient, S can also be expressed as

Example: Find the area of the surface that is generated by revolving the portion ofThe curve between x=0 and x=1 about the x-axis. 3y x

Solution: Since , we have and hence the surface area S is 3y x 2/ 3 ,dy dx x

1 2

0

1 3 2 2

0

1 3 4 1/2

0

10 1/2

0

103/2 3/2

1

2 1 ( )

2 1 (3 )

2 (1 9 )

2

36

2 2(10 1) 3.56

36 3 27u

dyS y dx

dx

x x dx

x x dx

u du

u