aula 26 calculo iialuno
TRANSCRIPT
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Professor: Carlos Alberto de Albuquerque
Blog: http://professorcarlosaa.blogspot.com.br/
Email:
CÁLCULO DIFERENCIAL E INTEGRAL II
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AULA
26
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COMPRIMENTO DE CURVAS PLANAS
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COMPRIMENTO DE CURVAS PLANAS
Aqui pressupomos que as funções f e g
tenham derivadas contínuas no intervalo
[a,b] e que essas derivadas não sejam
simultaneamente nulas.
Tais funções são chamadas continuamente
deriváveis e a curva C definida por elas de
curva lisa.
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COMPRIMENTO DE CURVAS PLANAS
Uma curva lisa C não se dobra para trás
nem muda de direção no intervalo [a,b], pois
ao longo de todo o intervalo.
02'2' gf
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COMPRIMENTO DE CURVAS PLANAS
Se x = f(t) e y = g(t), usando a notação de
Leibniz temos o seguinte resultado para o
comprimento do arco:
dtdt
dy
dt
dxL
b
a
22
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EXEMPLO 1A circunferência de um círculo
Determine o comprimento do círculo de raio r
definido parametricamente por
20
,cos
t
tsenryetrx
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SOLUÇÃO
Quando t varia de 0 para 2π, o círculo é
atravessado exatamente uma vez: logo, a
circunferência édt
dt
dy
dt
dxL
b
a
22
Determinamos
2222
22
cos
cos,
rttsenrdt
dy
dt
dx
e
trdt
dytsenr
dt
dx
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SOLUÇÃO
Logo
2
0
2
0
2
22
2 rtrdtr
dtdt
dy
dt
dxL
b
a
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EXEMPLO 2
Determine o comprimento
do astróide:
.20
,,cos 33
t
tsenytx
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SOLUÇÃO
Devido à simetria da curva em relação aos
eixos das coordenadas, seu comprimento é
quatro vezes o comprimento da porção que está
no primeiro quadrante.
Temos então:
ttsenttsendt
dy
tsenttsentdt
dx
tsenytx
2422
2
2422
2
33
cos9cos3
cos9cos3
,cos
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SOLUÇÃO
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SOLUÇÃO
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EXERCÍCIO 1
Determine o comprimento da curva
.13
2,32,1 ttytx
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SOLUÇÃO
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EXERCÍCIO 2
Determine o comprimento da curva
.30,2
3,
23 t
tytx
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SOLUÇÃO
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COMPRIMENTO DE CURVAS PLANAS
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COMPRIMENTO DE CURVAS PLANAS
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COMPRIMENTO DE CURVAS PLANAS
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COMPRIMENTO DE CURVAS PLANAS
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SOLUÇÃO
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SOLUÇÃO
222
22
22
22
2
24
2
4
24
4
24
24
111
xxxx
xx
xx
xx
eeee
ee
ee
eedx
dy
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SOLUÇÃO
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EXERCÍCIO 3
Determine o comprimento da curva
.30,23
12
32 xaxdexy
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SOLUÇÃO
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COMPRIMENTO DE CURVAS PLANAS
DEFINIÇÃO
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COMPRIMENTO DE CURVAS PLANAS
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SOLUÇÃO
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SOLUÇÃO
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COMPRIMENTO DE CURVAS PLANAS
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SOLUÇÃO
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EXERCÍCIO 4
Determine o comprimento da curva
.314
1
3
3
yayde
y
yx
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SOLUÇÃO
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FIM
DA AULA
26