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Calculo Diferencial e Integral II
Integrales Definidas Ciclo escolar 2013-2014
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Integrales Definidas
• La siguiente notación se lee:
La Integral definida de “𝒂” a “𝒃” de 𝒇(𝒙).
Y representa el área con signo de la región limitada por el eje 𝑋, la curva 𝑦 = 𝑓(𝑥)
y las rectas
𝑥 = 𝑎, 𝑥 = 𝑏.
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Ejemplos 2𝑥 − 1 𝑑𝑥
3
1
2 − 𝑥 𝑑𝑥6
2
𝑥
2+ 1 𝑑𝑥
5
1
𝑥
3− 1 𝑑𝑥
5
−1
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Sumas de Riemann
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Sumas de Riemann • Si hacemos 𝑎 = 𝑥0 < 𝑥1 < ⋯ < 𝑥𝑛−1 < 𝑥𝑛 = 𝑏, una
partición del segmento 𝑎 ≤ 𝑥 ≤ 𝑏, y Δ𝑥𝑗 = 𝑥𝑗+1 − 𝑥𝑗, entonces
𝑓 𝑥 𝑑𝑥𝑏
𝑎
= lim𝑛→∞
𝑓 𝑥𝑘 Δ𝑥𝑘
𝑛−1
𝑘=0
= lim𝑛→∞
𝑓 𝑥𝑘 Δ𝑥𝑘−1
𝑛
𝑘=1
• A esta expresión se le conoce como sumas de Riemann.
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Sumas de Riemann x fx dx fxdx
0 0.000 0.1
0.1 0.095 0.1 0.0095
0.2 0.180 0.1 0.0180
0.3 0.255 0.1 0.0255
0.4 0.320 0.1 0.0320
0.5 0.375 0.1 0.0375
0.6 0.420 0.1 0.0420
0.7 0.455 0.1 0.0455
0.8 0.480 0.1 0.0480
0.9 0.495 0.1 0.0495
1 0.500 0.1 0.0500
1.1 0.495 0.1 0.0495
1.2 0.480 0.1 0.0480
1.3 0.455 0.1 0.0455
1.4 0.420 0.1 0.0420
1.5 0.375 0.1 0.0375
1.6 0.320 0.1 0.0320
1.7 0.255 0.1 0.0255
1.8 0.180 0.1 0.0180
1.9 0.095 0.1 0.0095
2 0.000 0.1
0.665
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Sumas de Riemann
𝑥3𝑑𝑥2
1
2𝑥2 − 10 𝑑𝑥4
2
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Teorema Fundamental del Calculo
• Si 𝑓 𝑥 es continua en el intervalo 𝑎 ≤ 𝑥 ≤ 𝑏, y 𝐹 𝑥 es una antiderivada de 𝑓 𝑥 , entonces
𝑓 𝑥 𝑑𝑥𝑏
𝑎
= 𝐹 𝑥 𝑎
𝑏= 𝐹 𝑏 − 𝐹 𝑎
• teorema
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Ejemplos
𝑥2𝑑𝑥4
1
=𝑥3
3 1
4
=43
3−
13
3
=64
3−
1
3=
63
3= 21
𝑥4 − 2𝑥 + 1 𝑑𝑥4
2
=𝑥5
5− 𝑥2 + 𝑥
2
4
=4 5
5− 4 2 + 4
−2 5
5− 2 2 + 2
=964
5−
22
5=
942
5
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Ejemplos
2𝑥 − 3 4𝑑𝑥4
0
=1
2⋅
2𝑥 − 3 5
5 0
4
=2𝑥 − 3 5
10 0
4
=2 4 − 3 5
10
−2 0 − 3 5
10
=5 5
10−
−3 5
10
=3125
10−
−243
10=
3368
10
𝑥 cos 𝑥 𝑑𝑥𝜋/2
0
= 𝑥 sen 𝑥 + cos 𝑥 0
𝜋/2
=𝜋
2⋅ sen
𝜋
2+ cos
𝜋
2− 0 ⋅ sen 0 + cos 0
=𝜋
2⋅ 1 + 0 − 0 ⋅ 0 + 1
=𝜋
2− 1 =
𝜋
2− 1