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Calcular el área del cardiode
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Calcular el área del cardiode
1.
a) Calcular el
área del cardiode: 𝑟 = 𝑎(1 − 𝑐𝑜𝑠𝜃). Representar gráficamente. Trabajamos en y positivo 𝐴𝑅𝐸𝐴 = ∬ 𝑑𝐴 𝐷 𝑅 = {(𝑟, 𝜃) / 0 ≤ 𝜃 ≤ 𝜋 , 0 ≤ 𝑟 ≤ 𝑎(1 − 𝑐𝑜𝑠𝜃)} 𝐴𝑅𝐸𝐴 = 2 ∫ ∫ 𝑟𝑑𝑟𝑑𝜃 𝑎(1−𝑐𝑜𝑠𝜃) 0 𝜋 0
2.
𝐴𝑅𝐸𝐴 = 2
∫ [ 𝑟2 2 ] 𝑎(1 − 𝑐𝑜𝑠𝜃) 0 𝜋 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2 ∫ 𝑎2 𝜋 0 (1 − 𝑐𝑜𝑠𝜃)2 2 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2𝑎2 2 ∫(1 − 𝑐𝑜𝑠𝜃)2 𝜋 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 𝑎2 ∫ 1 − 2𝑐𝑜𝑠𝜃 + 𝑐𝑜𝑠2 𝜃 𝜋 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 𝑎2 ∫ 1 − 2𝑐𝑜𝑠𝜃 + 1 2 + 𝑐𝑜𝑠2𝜃 2 𝜋 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 𝑎2 [ 3 2 𝜃 − 2𝑠𝑒𝑛𝜃 + 𝑠𝑒𝑛2𝜃 4 ] 𝜋 0 𝐴𝑅𝐸𝐴 = 𝑎2 [ 3 2 𝜋 − 0 + 0 − 0 − 0 + 0] 𝐴𝑅𝐸𝐴 = 3𝜋𝑎2 2 b) Calcular el área comprendida entre los círculos: 𝑟 = 𝑎𝑐𝑜𝑠𝜃; 𝑟 = 𝑏𝑠𝑒𝑛𝜃 𝑐𝑜𝑛 𝑏 > 𝑎. c) Calcular el área de la lemniscata de Bernoulli 𝑟2 = 𝑎2 𝑐𝑜𝑠2𝜃.
3.
𝐴𝑅𝐸𝐴 = ∬
𝑑𝐴 𝐷 𝑅 = {(𝑟, 𝜃)/ 0 ≤ 𝜃 ≤ 𝜋 4 , 0 ≤ 𝑟 ≤ 𝑎√𝑐𝑜𝑠2𝜃} 𝐴𝑅𝐸𝐴 = 4 ∫ ∫ 𝑟𝑑𝑟𝑑𝜃 𝑎√𝑐𝑜𝑠2𝜃 0 𝜋 4 0 𝐴𝑅𝐸𝐴 = 4 ∫ [ 𝑟2 2 ] 𝑎√𝑐𝑜𝑠2𝜃 0 𝜋 4 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2 ∫[𝑎√𝑐𝑜𝑠2𝜃] 2 𝜋 4 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2𝑎2 ∫ 𝑐𝑜𝑠2𝜃 𝜋 4 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2𝑎2 [ 𝑠𝑒𝑛2𝜃 2 ] 𝜋 4 0 𝐴𝑅𝐸𝐴 = 𝑎2 d) Calcular el área interior a la curva 𝑟 = 2𝑠𝑒𝑛3𝜃 y exterior a 𝑟 = 1 en el primer cuadrante.
4.
𝐴𝑅𝐸𝐴 = ∬
𝑑𝐴 𝐷 𝑅 = {(𝑟, 𝜃)/ 0 ≤ 𝜃 ≤ 𝜋 3 , 0 ≤ 𝑟 ≤ 2𝑠𝑒𝑛3𝜃} 𝐴𝑅𝐸𝐴 = ∫ ∫ 𝑟𝑑𝑟𝑑𝜃 2𝑠𝑒𝑛3𝜃 0 𝜋 3 0 𝐴𝑅𝐸𝐴 = ∫ [ 𝑟2 2 ] 2𝑠𝑒𝑛3𝜃 0 𝜋 3 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = ∫ [ (2𝑠𝑒𝑛3𝜃)2 2 ] 𝜋 3 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = ∫ 2𝑠𝑒𝑛2 3𝜃 𝜋 3 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2 ∫ 1 2 − 𝑐𝑜𝑠6𝜃 2 𝜋 3 0 𝑑𝜃 𝐴𝑅𝐸𝐴 = 2 [ 𝜃 2 − 𝑠𝑒𝑛6𝜃 12 ] 𝜋 3 0 𝐴𝑅𝐸𝐴 = 2𝜋 6 = 𝜋 3
5.
e) Calcular el
volumen de la esfera de ecuación: 𝑥2 + 𝑦2 + 𝑧2 = 𝑎2 . a a
6.
𝑉𝑂𝐿𝑈𝑀𝐸𝑁 = ∬
𝑑𝐴 𝐷 𝑅 = {(𝑟, 𝜃)/ 0 ≤ 𝜃 ≤ 2𝜋 , 0 ≤ 𝑟 ≤ 𝑎} 𝑥2 + 𝑦2 + 𝑧2 = 𝑎2 𝑧 = √𝑎2 − 𝑥2 − 𝑦2 = √𝑎2 − 𝑟2 𝑉 = 2 ∫ ∫ √𝑎2 − 𝑟2𝑟𝑑𝑟𝑑𝜃 𝑎 0 𝑐𝑎𝑚𝑏𝑖𝑜 𝑑𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 {𝑢 = 𝑎2 − 𝑟2 𝑑𝑢 = −2𝑟𝑑𝑟 2𝜋 0 𝑉 = 2 ∫ [ √𝑢 −2 ] 𝑎 0 𝑑𝑢 2𝜋 0 𝑑𝜃 𝑉 = − ∫ [ 2𝑢 3 2 3 ] 𝑎 0 2𝜋 0 𝑑𝜃 = − ∫ [ 2(𝑎2 − 𝑟2) 3 2 3 ] 𝑎 0 2𝜋 0 𝑑𝜃 𝑉 = − 2 3 ∫ [(𝑎2 − 𝑟2 ) 3 2 ] 𝑎 0 2𝜋 0 𝑑𝜃 𝑉 = − 2 3 ∫ 0 − 𝑎3 2𝜋 0 𝑑𝜃 𝑉 = 2 3 𝑎3 [𝜃] 2𝜋 0 𝑉 = 2 3 𝑎3 2𝜋 𝑉 = 4𝜋 3 𝑎3 f) Calcular el volumen del solido contenido en el primer octante y acotado por las superficies 𝑧 = 𝑟 y el cilindro 𝑟 = 3𝑠𝑒𝑛𝜃.
7.
𝑉𝑂𝐿𝑈𝑀𝐸𝑁 = ∬
𝑑𝐴 𝐷 𝑅 = {(𝑟, 𝜃)/ 0 ≤ 𝜃 ≤ 𝜋 2 , 0 ≤ 𝑟 ≤ 3𝑠𝑒𝑛𝜃} 𝑧 = 𝑟 y el cilindro 𝑟 = 3𝑠𝑒𝑛𝜃.
8.
𝑉 = ∫
∫ (𝑟 − 0)𝑟𝑑𝑟𝑑𝜃 3𝑠𝑒𝑛𝜃 0 𝜋/2 0 𝑉 = ∫ ∫ (𝑟 − 0)𝑟𝑑𝑟𝑑𝜃 3𝑠𝑒𝑛𝜃 0 𝜋/2 0 𝑉 = ∫ ∫ 𝑟2 𝑑𝑟𝑑𝜃 3𝑠𝑒𝑛𝜃 0 𝜋/2 0 𝑉 = ∫ 27𝑠𝑒𝑛3 𝜃 3 𝑑𝜃 𝜋/2 0 𝑉 = 9 ∫ 𝑠𝑒𝑛𝜃(1 − 𝑐𝑜𝑠2 𝜃)𝑑𝜃 𝜋/2 0 𝑉 = 9 ∫ (𝑠𝑒𝑛𝜃 − 𝑠𝑒𝑛𝜃𝑐𝑜𝑠2 𝜃)𝑑𝜃 𝑐𝑎𝑚𝑏𝑖𝑜 𝑑𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 { 𝑢 = 𝑐𝑜𝑠𝜃 𝑑𝑢 = −𝑠𝑒𝑛𝜃 𝜋/2 0 𝑉 = 9 ∫ 𝑠𝑒𝑛𝜃𝑑𝜃 − 9 ∫ 𝑠𝑒𝑛𝜃𝑐𝑜𝑠2 𝜃 𝜋/2 0 𝑑𝜃 𝜋/2 0 𝑉 = 9 ∫ 𝑠𝑒𝑛𝜃𝑑𝜃 + 9 ∫ 𝑢2 𝜋/2 0 𝑑𝜃 𝜋/2 0 𝑉 = 9 [ −cos𝜃 + 𝑐𝑜𝑠3 𝜃 3 ] 𝜋 2 0 𝑉 = 9 (0 + 0 + 1 − 1 3 ) 𝑉 = 6𝑢3 g) Calcular el volumen del solido limitado por los cilindros de ecuaciones: 𝑥2 + 𝑦2 = 𝑎2 ; 𝑥2 + 𝑦2 = (𝑎 − 1)2 ; por el plano 𝑧 = 1 y por los planos coordenados.
9.
10.
𝑉𝑂𝐿𝑈𝑀𝐸𝑁 = ∬
𝑑𝐴 𝐷 𝑅 = {(𝑟, 𝜃)/ 0 ≤ 𝜃 ≤ 2𝜋 , 𝑎 − 1 ≤ 𝑟 ≤ 𝑎} 𝑥2 + 𝑦2 = 𝑎2 ; 𝑥2 + 𝑦2 = (𝑎 − 1)2 ; por el plano 𝑧 = 1 y z=0 𝑉 = ∫ ∫ (1 − 0)𝑟𝑑𝑟𝑑𝜃 𝑎 𝑎−1 2𝜋 0 𝑉 = 1 2 ∫ [𝑟2 ] 𝑎 𝑎 − 1 𝑑𝜃 2𝜋 0 𝑉 = 1 2 ∫ 𝑎2 −(𝑎 − 1)2 𝑑𝜃 2𝜋 0 𝑉 = 1 2 (𝑎2 −(𝑎 − 1)2) [𝜃] 𝜋 2 0 𝑉 = 2𝜋 2 (𝑎2 −(𝑎 − 1)2 ) 𝑉 = 𝜋(𝑎2 −(𝑎 − 1)2 ) 𝑉 = 𝜋(2𝑎 − 1)
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