Length of a Plane Curve - Arc Length - Classwork
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, "x $ "y
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lim
2
"x #0, "y #0
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% "y 2 ( 2 % / "y 2 2 ( 2
"x $ "y # '1 $ 2 * "x # '1 $ 1 4 * "x
'& 0 "x 3 *)
& "x )
2
2
So a chord length can be written as
,
% / "y 2 2 (
'1 $ 1 4 * "x 2 # ,
'& 0 "x 3 *)
% / "y 2 2 (
'1 $ 1 4 * "x
'& 0 "x 3 *)
"y
!!!!!!!!!!!! As "x approaches 0, you know that "x # f 5! x "
So the exact length of the curve between x # a and x # b is
b
L#
-
2
1 $ % f 5! x "& dx
NOTE : very few problems can be solved using the FTC.
a
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3
L#
-
1 $ 16 dx # 3 17
0
L # 9 $ 144 # 153 # 3 17
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2
-
L#
1 $ 25 dx # 2 26
0
3
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4
L#
-
1$
0
9x
dx
4
4 %/ 9 x 2
L # '11 $ 4
43
9 '&0
u # 1$
9x
9x
, du #
dx
4
4
!
"
32 4
( 8 10 10 . 1
* #
27
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!
3
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12
y 5 # t ! t 2 $ 2"
2
L#
-
2
-
1 $ t 2 ! t 2 $ 2" dt #
0
0
2
2
L#
t 4 $ 2 t $ 1 dt
- !t
2
0
% t 3 ( 14
ft
$ 1" dt # ' $ t* #
& 3 )0 3
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% !(
y # tan x
y # ex
%0, 2&
'&0, 4 *)
2
=?!!
5?!!!!
!4
L # - 1 $ e 2 x dx # 6.789
L
#
- 1 $ sec 4 x dx # 1.278
0
0
y # 4x . x2
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L#
1
%1, 4&
2
1 $ ! 4 . 2 x " dx # 6.126
y # x sin x
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2!
L#
-
%0, 2!&
2
1 $ ! x cos x $ sin x " dx # 15.065
0
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