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APLICACIONES DE LAS INTEGRALES
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RESUMEN DE : APLICACIONES DE LAS INTEGRALES
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TEMA 11: APLICACIONES DE LAS INTEGRALES Si se trata del área del recintodelimitado por dos curvas
1. ÁREA DE FIGURAS PLANAS
𝐶 ≔ {(𝑥, 𝑦) ∈ ℝ2 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓(𝑥) ≤ 𝑦 ≤ 𝑔(𝑥)}
DEFINICIÓN
el área será:
Sea 𝑓: [𝑎, 𝑏] → ℝ continua y ≥ 0 ∀𝑥 ∈ [𝑎, 𝑏]. El área del recinto
𝒃
𝐶 ≔ {(𝑥, 𝑦) ∈ ℝ2 : 𝑎 ≤ 𝑥 ≤ 𝑏, 0 ≤ 𝑦 ≤ 𝑓(𝑥)} 𝑓 (𝑥 ) Á𝒓𝒆𝒂(𝑪) = ∫ [𝒈(𝒙) − 𝒇(𝒙)] 𝒅𝒙
𝒂
viene dada por la integral:
𝑏
Á𝑟𝑒𝑎(𝐶) = ∫ 𝑓(𝑥) 𝑑𝑥
𝑎 En general, si las gráficas de ambas
funciones se cortan entre si varias veces,
el área del recinto C limitado por las
verticales 𝑥 = 𝑎, 𝑥 = 𝑏 y las curvas
Esta definición se puede extender a otros recintos planos. verticales 𝑓(𝑥) y 𝑔(𝑥) será:
DEFINICIÓN 𝒃
Á𝒓𝒆𝒂(𝑪) = ∫ |𝒇(𝒙) − 𝒈(𝒙)| 𝒅𝒙
Si la función fuese negativa, el área del recinto 𝒂
𝐶 ≔ {(𝑥, 𝑦) ∈ ℝ2 : 𝑎 ≤ 𝑥 ≤ 𝑏, 𝑓(𝑥) ≤ 𝑦 ≤ 0}
NOTA. Es fácil escribir las fórmulas análogas para áreas de regiones del tipo
sería:
{(𝑥, 𝑦) ∈ ℝ2 : 𝑐 ≤ 𝑦 ≤ 𝑑, 𝑔(𝑦) ≤ 𝑥 ≤ 𝑓(𝑦)}
𝑏
Á𝑟𝑒𝑎(𝐶) = − ∫ 𝑓(𝑥) 𝑑𝑥
𝑎
2. LONGITUD DE ARCOS DE CURVA
𝑓 (𝑥 ) DEFINICIÓN
Se define la longitud del arco de curva 𝑦 = 𝑓(𝑥) entre los puntos 𝐴(𝑎, 𝑓(𝑎)) y
𝐵(𝑏, 𝑓(𝑏)) como
En general, si la función no tiene signo 𝑏
2
constante, el área del recinto C sería la 𝑙 = ∫ √1 + (𝑓 ′(𝑥)) 𝑑𝑥
𝑎
suma de las áreas parciales de los recintos
donde se conserva el signo, 𝑓 (𝑥 )
o equivalentemente,
𝒃
Á𝒓𝒆𝒂(𝑪) = ∫ |𝒇(𝒙)| 𝒅𝒙
𝒂
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