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.
