Golden Spike National Historic Site, Promontory, Utah

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999

7.4 Day 1 Lengths of Curves

If we want to approximate the length of a curve, over a short distance we could measure a straight line.

ds

dx

dy

By the pythagorean theorem:

2 2 2ds dx dy

2 2ds dx dy

2 2ds dx dy We need to get dx out from under the radical.

2 22

2 2

dx dyS dxdx dx

2

21 dyL dxdx

2

1 b

a

dyL dxdx

Length of Curve (Cartesian)

Lengths of Curves:

2 9y x

0 3x

Example: 2 9y x

2dy xdx

23

01 dyL dx

dx

3 2

01 2 L x dx

3 2

01 4 L x dx

Now what? This doesn’t fit any formula, and we started with a pretty simple example!

9.74708875861L

The TI-89 gets:

2 9y x

0 3x

Example:

2 2 29 3 C 281 9 C

290 C

9.49C

The curve should be a little longer than the straight line, so our answer seems reasonable.

If we check the length of a straight line:

9.74708875861L

Example:

2 2 1x y

2 21y x 21y x

21

11 dyL dx

dx

3.1415926536

21

dy xdx x

Example:Find the arc length of the graph of

3 1 1 ,2 .6 2 2xy on

x

Example:Find the arc length of the graph of

3 1 1 ,2 .6 2 2xy on

x

22

1 12

dy xdx x

Solution:

222

21 2

1 112

s x dxx

22

21 2

1 12x dx

x

23

1 2

1 12 3x

x

1 13 472 6 24

3316

Example:Find the arc length of the graph of ln cos 0, .

4y x on

Example:Find the arc length of the graph of ln cos 0, .

4y x on

tandy xdx

Solution:

42

0

1 tans x dx

4

0ln sec tanx x

ln 2 1 ln1 4

2

0

secs x dx

4

0

secs x dx

0.881.

Homework:7.4 pg. 483 #1-19 odd.