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UNIVERSIDAD PEDAGÓGICA
NACIONAL FRANCISCO
MORAZÁN
TEMA: APLICACIÓN DE LAS
PROPIEDADES DE LA INTEGRAL
DEFINIDA
GRADO: ÚNDÉCIMO
TEGUCIGALPA M.D.C.
PRÁCTICA PROFESIONAL II
Propiedades
II
Ejemplo
Propiedades
I
1
EQUIPO 4
Recordemos las propiedades de la integral definida:
1. 𝑎
𝑏
𝑘 𝑓 𝑥 𝑑𝑥 = 𝑘 𝑎
𝑏
𝑓 𝑥 𝑑𝑥
2. 𝑎
𝑏
𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 = 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 + 𝑎
𝑏
𝑔 𝑥 𝑑𝑥
3. 𝑎
𝑏
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 = 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 − 𝑎
𝑏
𝑔 𝑥 𝑑𝑥
Propiedades
II
Ejemplo
Propiedades
I
3
2
Aplicaremos las propiedades de la integral definida para
resolver el siguiente ejercicio:
1
2
(2𝑥3
− 3𝑥 + 1)𝑑𝑥
= 2
1
2
𝑥3
𝑑𝑥 − 3
1
2
𝑥 𝑑𝑥 +
1
2
𝑑𝑥
= 2 ·
1
4
𝑥4
1
2
− 3 ·
1
2
𝑥2
1
2
+ 𝑥
1
2
=
1
2
(24
− 14
) −
3
2
(22
− 12
) + (2 − 1)
=
1
2
(16 − 1) −
3
2
(4 − 1) + 1
=
15
2
−
9
2
+ 1
= 4
1
2
(2𝑥3
− 3𝑥 + 1)𝑑𝑥 = 4
1. 𝑎
𝑏
𝑘 𝑓 𝑥 𝑑𝑥 = 𝑘 𝑎
𝑏
𝑓 𝑥 𝑑𝑥
2. 𝑎
𝑏
𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 = 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 + 𝑎
𝑏
𝑔 𝑥 𝑑𝑥
3. 𝑎
𝑏
𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 = 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 − 𝑎
𝑏
𝑔 𝑥 𝑑𝑥
Propiedades
II
Ejemplo
Propiedades
I
5
4
Otras propiedades muy útiles son las siguientes:
1. 𝑎
𝑎
𝑓 𝑥 𝑑𝑥 = 0
2. 𝑏
𝑎
𝑓 𝑥 𝑑𝑥 = − 𝑎
𝑏
𝑓 𝑥 𝑑𝑥
3. 𝑎
𝑏
𝑓(𝑥)𝑑𝑥 + 𝑏
𝑐
𝑓 𝑥 𝑑𝑥 = 𝑎
𝑐
𝑓 𝑥 𝑑𝑥
Propiedades
II
Ejemplo
Propiedades
I
7
6
Aplicaremos las propiedades de la integral definida para
resolver el siguiente ejercicio:
0
2
−𝑥2
+ 4𝑥 + 2 𝑑𝑥 +
2
3
−𝑥2
+ 4𝑥 + 2 𝑑𝑥
=
0
3
−𝑥2
+ 4𝑥 + 2 𝑑𝑥
= −1 ·
1
3
𝑥3
0
3
+ 4 ·
1
2
𝑥2
0
3
+ 2𝑥
0
3
= −
1
3
33
− 03
+ 2 (32
− 02
) + (2 · 3 − 2 · 0)
= −
1
3
(27) − 2 (9) + 6
= −21
0
3
−𝑥2
+ 4𝑥 + 2 𝑑𝑥 = −21
1. 𝑎
𝑎
𝑓 𝑥 𝑑𝑥 = 0
2. 𝑏
𝑎
𝑓 𝑥 𝑑𝑥 = − 𝑎
𝑏
𝑓 𝑥 𝑑𝑥
3. 𝑎
𝑏
𝑓(𝑥)𝑑𝑥 + 𝑏
𝑐
𝑓 𝑥 𝑑𝑥 = 𝑎
𝑐
𝑓 𝑥 𝑑𝑥
Propiedades
II
Ejemplo
Tema
1
8
Las matemáticas son la creación más poderosa y
bella del espíritu humano. –
Stefan Banach
ESTE RECURSO DIDÁCTICO
LLEGA A USTEDES GRACIAS A
LA UNIVERSIDAD
PEDAGÓGICA NACIONAL
“FRANCISCO MORAZÁN”
Aplicación propiedades integral

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Aplicación propiedades integral

  • 1.
  • 2. UNIVERSIDAD PEDAGÓGICA NACIONAL FRANCISCO MORAZÁN TEMA: APLICACIÓN DE LAS PROPIEDADES DE LA INTEGRAL DEFINIDA GRADO: ÚNDÉCIMO TEGUCIGALPA M.D.C. PRÁCTICA PROFESIONAL II
  • 3. Propiedades II Ejemplo Propiedades I 1 EQUIPO 4 Recordemos las propiedades de la integral definida: 1. 𝑎 𝑏 𝑘 𝑓 𝑥 𝑑𝑥 = 𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 2. 𝑎 𝑏 𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 + 𝑎 𝑏 𝑔 𝑥 𝑑𝑥 3. 𝑎 𝑏 𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 − 𝑎 𝑏 𝑔 𝑥 𝑑𝑥
  • 4. Propiedades II Ejemplo Propiedades I 3 2 Aplicaremos las propiedades de la integral definida para resolver el siguiente ejercicio: 1 2 (2𝑥3 − 3𝑥 + 1)𝑑𝑥 = 2 1 2 𝑥3 𝑑𝑥 − 3 1 2 𝑥 𝑑𝑥 + 1 2 𝑑𝑥 = 2 · 1 4 𝑥4 1 2 − 3 · 1 2 𝑥2 1 2 + 𝑥 1 2 = 1 2 (24 − 14 ) − 3 2 (22 − 12 ) + (2 − 1) = 1 2 (16 − 1) − 3 2 (4 − 1) + 1 = 15 2 − 9 2 + 1 = 4 1 2 (2𝑥3 − 3𝑥 + 1)𝑑𝑥 = 4 1. 𝑎 𝑏 𝑘 𝑓 𝑥 𝑑𝑥 = 𝑘 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 2. 𝑎 𝑏 𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 + 𝑎 𝑏 𝑔 𝑥 𝑑𝑥 3. 𝑎 𝑏 𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 − 𝑎 𝑏 𝑔 𝑥 𝑑𝑥
  • 5. Propiedades II Ejemplo Propiedades I 5 4 Otras propiedades muy útiles son las siguientes: 1. 𝑎 𝑎 𝑓 𝑥 𝑑𝑥 = 0 2. 𝑏 𝑎 𝑓 𝑥 𝑑𝑥 = − 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 3. 𝑎 𝑏 𝑓(𝑥)𝑑𝑥 + 𝑏 𝑐 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑐 𝑓 𝑥 𝑑𝑥
  • 6. Propiedades II Ejemplo Propiedades I 7 6 Aplicaremos las propiedades de la integral definida para resolver el siguiente ejercicio: 0 2 −𝑥2 + 4𝑥 + 2 𝑑𝑥 + 2 3 −𝑥2 + 4𝑥 + 2 𝑑𝑥 = 0 3 −𝑥2 + 4𝑥 + 2 𝑑𝑥 = −1 · 1 3 𝑥3 0 3 + 4 · 1 2 𝑥2 0 3 + 2𝑥 0 3 = − 1 3 33 − 03 + 2 (32 − 02 ) + (2 · 3 − 2 · 0) = − 1 3 (27) − 2 (9) + 6 = −21 0 3 −𝑥2 + 4𝑥 + 2 𝑑𝑥 = −21 1. 𝑎 𝑎 𝑓 𝑥 𝑑𝑥 = 0 2. 𝑏 𝑎 𝑓 𝑥 𝑑𝑥 = − 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 3. 𝑎 𝑏 𝑓(𝑥)𝑑𝑥 + 𝑏 𝑐 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑐 𝑓 𝑥 𝑑𝑥
  • 7. Propiedades II Ejemplo Tema 1 8 Las matemáticas son la creación más poderosa y bella del espíritu humano. – Stefan Banach
  • 8. ESTE RECURSO DIDÁCTICO LLEGA A USTEDES GRACIAS A LA UNIVERSIDAD PEDAGÓGICA NACIONAL “FRANCISCO MORAZÁN”