Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland,...
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Transcript of Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland,...
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.