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calculo2_a


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\uf072 \uf03d 1 \uf02d 2sin\uf071 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
b) Se b \uf03c a \uf0de a curva não tem laço 
\uf072 \uf03d 3 \uf02b 2cos\uf071 \uf072 \uf03d 3 \uf02d 2cos\uf071 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-12
\uf072 \uf03d 3 \uf02b 2sin\uf071 \uf072 \uf03d 3 \uf02d 2sin\uf071 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2.3.6 Cardióides 
São limaçons onde a \uf03d b. 
\uf072 \uf03d a\uf0d7( 1 \uf0b1 cos\uf071) ou \uf072 \uf03d a\uf0d7( 1 \uf0b1 sin\uf071), onde a \uf0ce R. 
\uf072 \uf03d 2(1 \uf02b cos\uf071) \uf072 \uf03d 2(1 \uf02d cos\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
\uf072 \uf03d 2(1 \uf02b sin\uf071) \uf072 \uf03d 2(1 \uf02d sin\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2.3.7 Lemniscata de Bernoulli 2.3.8 Espiral de Arquimedes 
\uf0722 \uf03d a2\uf0d7cos(2\uf071), onde a \uf0ce R. \uf072 \uf03d a\uf0d7\uf071, onde a \uf03e 0. 
\uf0722 \uf03d 4\uf0d7cos(2\uf071) \uf072 \uf03d \uf071 (Obs: 0 \uf0a3 \uf071 \uf0a3 4\uf070) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-13
2.3.9 Rosáceas 
\uf072 \uf03d a\uf0d7cos(n\uf0d7\uf071) ou \uf072 \uf03d a\uf0d7sin(n\uf0d7\uf071), onde a \uf0ce R e n \uf0ce N. 
\uf072 \uf03d 3\uf0d7cos(2\uf071) \uf072 \uf03d 3\uf0d7sin(2\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
\uf072 \uf03d 4\uf0d7cos(3\uf071) \uf072 \uf03d 4\uf0d7sin(3\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
\uf072 \uf03d 4\uf0d7cos(4\uf071) \uf072 \uf03d 4\uf0d7sin(4\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
\uf072 \uf03d 4\uf0d7cos(5\uf071) \uf072 \uf03d 4\uf0d7sin(5\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
\uf072 \uf03d 4\uf0d7cos(6\uf071) \uf072 \uf03d 4\uf0d7sin(6\uf071) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 Cálculo II \u2013 (Lauro / Nunes) 2-14
2.4 Áreas em Coordenadas Polares 
Vamos iniciar determinando a área em um setor circular e depois, desenvolver para 
coordenadas polares. 
2.4.1 Área de um Setor Circular 
Área de um setor circular de raio r e abertura \uf044\uf071 que será calculada através de uma 
regra de três simples: 
\uf044\uf071
Setor
\uf072
 
Área Total (At) \uf03d \uf070\uf0722 
Área Setor (As)\uf03d ? 
At \u2013 2\uf070 \uf070\uf0722 \u2013 2\uf070 
As \u2013 \uf044\uf071 As \u2013 \uf044\uf071 
 As \uf03d 
\uf070
\uf071\uf044\uf0d7\uf070\uf072
2
2
 \uf03d 
2
2 \uf071\uf044\uf0d7\uf072 
 As \uf03d 
2
1
\uf0722\uf044\uf071 
2.4.2 Áreas em Coordenadas Polares (dedução) 
Seja f uma função contínua e não-negativa no intervalo fechado [\uf061 , \uf062]. Seja R uma 
região limitada pela curva cuja equação é \uf072 \uf03d f(\uf071) e pelas retas \uf071 \uf03d \uf061 e \uf071 \uf03d \uf062. Então, a região 
R é a que está mostrada na figura seguinte. 
\uf071
f \uf062
\uf061
\uf072
\uf062\uf03d
\uf071\uf03d\uf061
\uf03d
R
( )\uf071
O 
Considere uma partição \uf044 de [\uf061 , \uf062] definida por: 
\uf061 \uf03d \uf0710 \uf03c \uf0711 \uf03c \uf0712 \uf03c \uf0bc \uf03c \uf071i\uf02d1 \uf03c \uf071i \uf03c \uf071i\uf02b1 \uf03c \uf0bc \uf03c \uf071n\uf02d1 \uf03c \uf071n \uf03d \uf062. 
Desta forma, definimos n subintervalos do tipo [\uf071i\uf02d1 , \uf071i], onde i \uf03d 1, 2, \uf0bc, n. 
\uf071
f\uf072
\uf062\uf03d
\uf071\uf03d\uf061
\uf03d ( )\uf071
O
\uf044i\uf071\uf03d \uf071i \uf071i\uf02d \uf02d1\uf071i
\uf071i\uf02d1
( )
 
A medida em radianos do ângulo entre as retas \uf071 \uf03d \uf071i\uf02d1 e \uf071 \uf03d \uf071i é denotada por \uf044i\uf071. 
Tome \uf078i como sendo um valor de \uf071 no i-ésimo subintervalo e considere f(\uf078i) o raio do 
setor circular neste subintervalo, como mostra a figura seguinte. 
 Cálculo II \u2013 (Lauro / Nunes) 2-15
\uf071
f\uf072
\uf062\uf03d
\uf071\uf03d\uf061
\uf03d ( )\uf071
O
\uf044i\uf071
\uf078i
\uf071i\uf02d1
f ( )Raio do setor
\uf078i
 
Como foi visto anteriormente, a área do setor é dada por: 
 \uf05b \uf05d \uf071\uf044\uf078 iif 2)(2
1 
Existe um setor circular para cada um dos n subintervalos. A soma das medidas das 
áreas é: 
 \uf05b \uf05d \uf071\uf044\uf078 121)(2
1 f \uf02b \uf05b \uf05d \uf071\uf044\uf078 222 )(2
1 f \uf02b\uf0bc\uf02b \uf05b \uf05d \uf071\uf044\uf078 iif 2)(2
1
\uf02b\uf0bc\uf02b \uf05b \uf05d \uf071\uf044\uf078 nnf 2)(2
1 
Que pode ser escrita através da somatória: 
 \uf05b \uf05d\uf0e5
\uf03d
\uf071\uf044\uf078
n
i
iif
1
2)(
2
1 
Tome A como a área da região R e seja \uf044 a norma da partição \uf044, isto é, \uf044 é o 
maior valor de \uf044i\uf071. Então a área é definida como: 
 A \uf03d \uf05b \uf05d\uf0e5
\uf03d
\uf0ae\uf044
\uf071\uf044\uf078
n
i
iif
1
2
0
)(
2
1lim 
Este limite é a seguinte integral definida: 
 A\uf03d \uf05b \uf05d\uf0f2
\uf062
\uf061
\uf071\uf071 df 221 )( 
Teorema 
Se f é contínua e f (\uf071) \uf0b3 0 em [\uf061, \uf062], onde 0 \uf0a3 \uf061 \uf03c \uf062 \uf0a3 2\uf070, então a área A da região 
delimitada pelos gráficos de \uf072 \uf03d f (\uf071), \uf071 \uf03d \uf061 e \uf071 \uf03d \uf062 é dada por: 
 A\uf03d \uf05b \uf05d\uf0f2
\uf062
\uf061
\uf071\uf071 df 221 )( \uf03d \uf0f2
\uf062
\uf061
\uf071\uf072 d221 
 
 
 
 
 
 
 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-16
Exemplos 
8. Calcule a área da região delimitada pela lemniscata de Bernoulli, de equação \uf0722\uf03d4 \uf0712cos . 
Resolução: 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Resposta: A = 4 u.a. 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-17
9. Calcular a área da região interna à rosácea \uf072 \uf03d \uf0712sina . 
Resolução: 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Resposta: A\uf03d
2
2a\uf070 u.a. 
 Cálculo II \u2013 (Lauro / Nunes) 2-18
10. Calcular a área da interseção das regiões limitadas pelas curvas \uf072\uf03d3 \uf071cos e \uf072\uf03d1+ \uf071cos . 
Resolução: 
Tipo de curva \uf071 0 
6
\uf070 
4
\uf070 
3
\uf070 
2
\uf070 
3
2\uf070 
4
3\uf070 
6
5\uf070 \uf070 
Circunferência 3 \uf071cos \uf072~ 
Cardióide 1+ \uf071cos \uf072~ 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-19
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Resposta: A\uf03d
4
5\uf070 u.a. 
11. Calcule a área da região limitada pela curva dada em coordenadas polares por \uf072 \uf03d \uf071tg , 
com 0 \uf0a3 \uf071 \uf03c 
2
\uf070 , pela