integrales region tipo 2
TRANSCRIPT
Cálculo integral
EjerciciosEjercicios
Calculo Calculo IntegralIntegral
Cálculo integral
EJERCICIO 1 REGIÓN TIPO I
f(x) g(x)y=3 x−x2 y=x−4
A (R I )=∫a
b
[ f ( x )−g(x )] dx
A (R I )= ∫−1.23607
3.23607
[ (3 x−x2 )−( x−4 ) ]dx
A (R I )= ∫−1.23607
3.23607
(3x−x2 )dx− ∫−1.23607
3.23607
( x−4 )dx
A (R I )=[ 32 x2− x3
3 ]−1.23607
3.23607
−[ x22 −4]−1.23607
3.23607
A (R I )=1.49069985−(−13.41642)
A (R I )=14.90711985u2
A(RI)
(3.23607, -0.76393)
(-1.23607, -5.23607)
A(RI)
(1.5, 2.25)
(3.23607, -0.76393)
(-1.23607, -5.23607)
Cálculo integral
EJERCICIO 1 REGIÓN TIPO II
f(x) g(x)
y=3 x−x2
94− y=x2−3 x+ 9
4
94− y=( x−32 )
2
√ 94− y=x−32x=32∓√ 94− y
y=x−4y+4=x
A1 (RII )=∫c
d
[ f 1( y)−f 2( y)] dy
A1 (RII )= ∫−0.76393
2.25 [( 32 +√ 94− y )−(32−√ 94− y )]dyA1 (RII )= ∫
−0.76393
2.25 [ 32+√ 94− y−32+√ 94− y ]dy
A1 (RII )= ∫−0.76393
2.25
2√ 94− y dy
A1 (RII )=[−43 ( 94− y )32 ] 2.25
−0.76393
A1 (RII )=[0−(−6.976514138)]
A1 (RII )=6.9765u2
A1(RII)
R
Cálculo integral
A2 (R II )=∫c
d
[ g( y )−f 2( y )] dy
A2 (R II )= ∫−5.23607
−0.76393 [ ( y+4 )−( 32−√ 94− y )]dyA2 (R II )= ∫
−5.23607
−0.76393 [ y+4−32 +√ 94− y ]dyA2 (R II )= ∫
−5.23607
−0.76393 [ y+√ 94− y+ 52 ]dyA2 (R II )=[ y22 −2
3 ( 94− y)32+ 52y ]−0 .76393−5 .23607
A2 (R II )=[−5.1062−(−13.0368)]
A2 (RII )=7.9306u2
A (RII )=A1 (RII )+A2 (R II )=14.9071u2
A2(RII)
Cálculo integral
EJERCICIO 2 REGIÓN TIPO I
f(x) g(x) y=−2x2 y=3 x−6
A (RI )=∫a
b
[ f ( x )−g(x )] dx
A (RI )= ∫−1.09545
1.09545
[ (−2x2 )−(3 x2−6 ) ]dx
A (RI )= ∫−1.09545
1.09545
[ (−5 x2+6 ) ]dx
A (RI )=¿[−53 x3+6 x]−1.095451.09545
A (RI )=4.38178046+4.38178046
A (RI )=8.76356092u2
A(RI) (1.09545, -2.4)(-1.09545, -2.4)
Cálculo integral
EJERCICIO 2 REGIÓN TIPO II
f(x) g(x) y=−2x2
x=±√− y2
y=3 x−6
x=±√ y+63
A1 (RII )=∫c
d
g1( y )dy
A1 (R II )=∫−6
0 [√ y+63 ]dyA1 (R II )=[2( y+63 )
32 ] 0−6
A1 (R II )=5.6568−0
A1 (R II )=5 .6568(2)
A1 (R II )=¿11.3137u2
A2 (RII )=∫c
d
[ g1( y )− f 1( y )]dy
A2 (RII )=∫−2.4
0 [√ y+63 −√− y2 ]dy
A2 (R II )=[2( y+63 )32+ 43 (− y2 )
32 ] 0
−2.4
A2 (R II )=5.6568−4.3817
A2 (R II )=1.2751(2)
A2 (R II )=2.5502u2
A (RI)
(0, -6)
(1.41421, 0)
(1.09545, -2.4)(-1.09545, -2.4)
(1.41421, 0)
A1(RI)
Cálculo integral
A (RII )=A1 (RII )−A2 (R II )=8.7635u2
Cálculo integral
EJERCICIO 3 REGIÓN TIPO I
f(x) g(x) y=−2x2 y=3 x2−6
A1 (R I )=∫a
b
[g (x)] dx
A1 (R I )= ∫−1.41421
1.414211
[3 x2−6 ]dy
A1 (R I )=[ x3−6 x ]−1.414211.41421
A1 (R I )=−5.6568−5.6568=−11.3137u2
A1 (R I )=11.3137u2
A2 (RI )=∫a
b
[ f ( x )−g(x )]dx
A2 (RI )= ∫−1.09545
1.09545
[ (−2x2 )−(3x2−6 ) ]dx
A2 (RI )= ∫−1.09545
1.09545
[ (−5x2+6 ) ]dx
A2 (RI )=[−53 x3+6 x]−1.09545
1.09545
A2 (RI )=4.38178046+4.38178046
A2 (RI )=8.7635u2
(1.41421, 0)(-1.41421, 0)
A2(RI) (1.09545, -2.4)(-1.09545, -2.4)
A1(RI)
Y el eje x
Cálculo integral
A (RI )=A1 (RI )−A2 (R I )≈2.5502u2
Cálculo integral
EJERCICIO 3 REGIÓN TIPO II
f(x) g(x) y=−2x2
x =±√− y2
y=3 x2−6
x = ±√ y+63
A (R II )=∫c
d
[ g1( y )−f 1( y )] dy
A (R II )=∫−2.4
0 [√ y+63 −√− y2 ]dy
A (R II )=[2( y+63 )32+ 43 (− y2 )
32 ] 0
−2.4
A (R II )=5.6568−4.3817
A (R II )=1.2751(2)
A (R II )≈2.5502u2
(1.09545, -2.4)(-1.09545, -2.4)
(1.41421, 0)
Y el eje x
Cálculo integral
EJERCICIO 4REGIÓN TIPO I
f(x) g(x) y=x2+6 y=2x3
A (RI )=∫a
b
[ f ( x )−g(x )] dx
A (RI )= ∫0
1.62964
[ (x2+6 )−(2x3 ) ]dx
A (RI )=[−x42 +x3
3+6 x ]
0
1.62964
A (RI )=7.6940−0
A (RI )≈7.6940u2
(1.62964, 8.65572)
(0, 0)
A(RI)
Y, el eje y
Cálculo integral
EJERCICIO 4REGIÓN TIPO II
f(x) g(x) y=x2+6
x=±√ y−6
y=2x3
x=3√ y2
A1 (RII )=∫c
d
[ g ( y ) ]dy
A1 (RII )=∫0
6 [ 3√ y2 ]dyA1 (RII )=[ 3
43√2y43 ]0
6
A1 (RII )≈6.4901u2
A2 (RII )=∫c
d
[g ( y )− f 1( y) ]dy
A2 (RII )=∫0
6 [ 3√ y2−√ y−6]dyA2 (RII )=[ 3
4 3√2y43−23
( y−6 )32 ]6
8.65572
A2 (RII )=7.6940−6.4901
A2 (RII )≈1.2039u2
(1.44225, 6)
(0, 0)
(1.44225, 6)
(0, 0)
(1.62964, 8.65572)
A1(RII)
Y, el eje y
Cálculo integral
A (RII )=A1 (RII )+A2 (R II )≈7.6940u2
Cálculo integral
EJERCICIO 5REGIÓN TIPO I
f(x) g(x) y=x2+6 y=2x3
A (R I )=∫a
b
[ f ( x )−g(x )] dx
A (R I )= ∫1.62964
3
[2x3−(x2+6 ) ]dx
A (R I )=[ x42 −x3
3−6 x ]
1.62964
3
A (R I )=13.5−(−7.6940)
A (R I )≈21.1940u2 (0, 0)
(1.62964, 8.65572)
(3, 54)
x = 3
Cálculo integral
EJERCICIO 5REGIÓN TIPO II
f(x) g(x) y=x2+6x1=√ y−6x2=√ y−6
y=2x3
x=( y2 )1/3
A1.1 (R II )=∫c
d
[3−f 1 ( y ) ]dy
A1.1 (R II )=∫6
15
[3−√ y−6 ] dy
A1.1 (R I )=[3 y−23 ( y−6 )32 ]6
15
A1.1 (R I )=27−18
A1.1 (R I )≈9u2
A1.2 (RII )=∫0
6
3dy
A1.2 (RI )=[3 y ]06
A1.2 (RI )=18u2
A1 (RI )=27u2
(0, 0)
(3, 15)
A1(RII)
x = 3
Cálculo integral
A2 (RII )=∫c
d
[3−g1 ( y ) ]dy
A2 (RII )=∫0
54
[3−( y2 )1 /3]dy
A2 (RI )=[3 y−32 ( y2 )43 ]0
54
A2 (RI )≈40.5u2
A3.1 (RII )=∫c
d
[ g ( y ) ] dy
A3.1 (RII )=∫0
6 [3√ y2 ]dyA3.1 (RII )=[ 34 3√2 y
43 ]0
6
A3.1 (RII )≈6.4901u2
A3.2 (RII )=∫c
d
[g ( y )−f 1( y )]dy
A3.2 (RII )=∫0
6 [3√ y2−√ y−6]dyA3.2 (RII )=[ 34 3√2 y
43−23
( y−6 )32 ]6
8.65572
A3.2 (RII )=7.6940−6.4901
A3.2 (RII )≈1.2039u2
A3 (RII )≈7.6940u2
(0, 0)
(1.62964, 8.65572)
(3, 54)
(0, 0)
A1(RII)
A3(RII)
Cálculo integral
A (RII )=A1 (RII )+A3 (RII )−A2 (R II )≈21.1940u2
Cálculo integral
EJERCICIO 6REGIÓN TIPO I
f(x) g(x) y=8−x2 y=x
A (RI )=∫a
b
[ f ( x )−g(x )] dx
A (RI )= ∫−3.37228
2.37228
[8−x2−x ] dx
A (RI )=[8 x− x33 −x2
2 ]−3.37228
2.37228
A (RI )=11.7142−(−19.8808)
A (RI )≈31.5950u2
(2.37228, 2.37228)
(-3.37228, -3.37228)
A(RI)
Cálculo integral
EJERCICIO 6REGIÓN TIPO II
f(x) g(x) y=8−x2
x=±√8− y
y=x
x= y
A1 (R II )=∫c
d
[ f 1( y)] dy
A1 (R II )= ∫2.37228
8
[ √8− y ] dy
A1 (R II )=[−23 (8− y )32 ]2.37228
8
A1 (R II )=8.9003u2
Multiplicar por 2 porque tiene un lado simétrico
A1 (R II )=8.9003(2)
A1 (RII )≈17.8007
A2 (RII )=∫c
d
[g ( y )− f 2( y) ]dy
A2 (RII )= ∫−3.37228
2.37228
[ y−(−√8− y ) ]dy
A2 (RII )=[ y22 −23
(8− y )32 ]
−3.37228
2.37228
A2 (RII )=−6.0865−(−19.8808)
A2 (RII )≈13.7943u2
(0, 8)
(2.37228, 2.37228)(-2.37228, 2.37228) 2.37228)
(3, 54)
(2.37228, 2.37228)
(-3.37228, -3.37228)
A1(RII)
Cálculo integral
A (RII )=A1 (RII )+A2 (R II )≈31.5950u2
Cálculo integral
EJERCICIO 7REGIÓN TIPO I
f(x) g(x) y=x+1 y= (x−1 )2
A (R I )=∫a
b
[ f ( x )−g(x )] dx
A (R I )=∫0
3
[ ( x+1 )−( x−1 )2 ]dx
A (R I )=∫0
3
[ x+1−x2+2x−1¿ ]dx
A (R I )=∫0
3
[3 x−x2 ]dx
A (R I )=[ 32 x2− x3
3 ]0
3
A (R I )=272
−9
A (R I )≈ 92u2
(3, 4)
(0, 1)A(RI)
Cálculo integral
EJERCICIO 7REGIÓN TIPO II
f(x) g(x) y=x+1
x= y−1
y= (x−1 )2
x=1∓√ y
A1 (RII )=∫c
d
[ g( y )−f 1( y )] dy
A1 (RII )=∫1
4
[1+√ y− y+1 ] dy
A1 (RII )=∫1
4
(− y+√ y+2 )dy
A1 (RII )=[− y22 + 2 y32
3+2 y ]
1
4
A1 (RII )=163
−136
A1 (RII )≈ 196u2
A2 (RII )=∫c
d
[ g1( y )−g2( y)] dy
A2 (RII )=∫0
1
[1+√ y−(1−√ y) ]dy
A2 (RII )=∫1
4
2√ y dy
A2 (RII )=[ 4 y32
3 ]0
1
A2 (RII )≈ 43u2
(1, 0)
(0, 1)
(3, 4)
(1, 0)
(0, 1)
(3, 4)
A1(RII)
Cálculo integral
A (RII )=A1 (RII )+A2 (R II )≈ 92u2
Cálculo integral
EJERCICIO 8REGIÓN TIPO I
f(x) g(x)x=( y−2 )2
y1=2+√xy2=2−√x
y=x−2
A1 (RI )=∫a
b
[ f 1 ( x )−g(x) ]dx
A1 (RI )= ∫0
6.56155
[(2+√x )−( x−2 ) ] dx
A1 (RI )= ∫0
6.56155
[4+√ x−x ]dx
A1 (RI )=[4 x+ 23 ( x )32− x
2
2 ]0
6.56155
A1 (RI )=15.9243−0¿
A1 (RI )≈15.9243u2
A2 (RI )=∫a
b
[ f 2 ( x )−g(x )]dx
A2 (RI )= ∫0
2.43845
[ (2−√x )−(x−2)]dx
A2 (RI )= ∫0
2.43845
[4−√x−x ] dx
A2 (RI )=[4 x−23 ( x )32− x
2
2 ]0
2.43845
A2 (RI )≈4.2422u2
(0,2)
(6.56155, 4.56155)
(0, -2)
(2.43845, 0.43845)
A1(RI)
A2(RI)
Cálculo integral
A (RI )=A1 (RI )−A2 (R I )≈11.6821u2
Cálculo integral
EJERCICIO 8REGIÓN TIPO II
f(x) g(x)x=( y−2 )2 y=x−2
x= y+2
A (RII )=∫c
d
[ g ( x )−f ( x ) ] dx
A (RII )= ∫0.43845
4.56155
[( y+2)− ( y−2 )2 ]dy
A (RII )= ∫0.43845
4.56155
[− y2+5 y−2 ] dy
A (RII )=[− y33 + 5 y2
2−2 y] 4.561550.43845
A (RII )=11.25773297−(−0.424399)¿
A (RII )≈11.6821u2
(6.56155, 4.56155)
(2.43845, 0.43845)
Cálculo integral
EJERCICIO 9REGIÓN TIPO I
f(x) g(x)y=x+1 y= (x−1 )2
A (RI )=∫a
b
[ f ( x )−g(x )] dx
A (RI )=∫0
3
[ ( x+1 )−( x−1 )2 ]dx
A (RI )=∫0
3
[−x2+3 x ] dx
A (RI )=[−x33 +32x2]
0
3
A (RI )=−9+ 272
A (RI )≈ 92u2
(1, 0)
(3, 4)
Y ejes coordenados
Cálculo integral
EJERCICIO 9REGIÓN TIPO II
f(x) g(x)y=x+1x= y−1
y= (x−1 )2
x1=1+√ yx2=1−√ y
A1 (RII )=∫c
d
[g1 ( y )−g2 ( y ) ]dy
A1 (RII )=∫0
1
[ ( y−1 )−(1−√ y ) ]dy
A1 (RII )=∫0
1
[2√ y ] dy
A1 (RII )=[ 43 y32 ]10
A1 (RII )≈ 43u2
A1 (RII )=∫c
d
[ f 1 ( y )−g ( y ) ]dy
A1 (RII )=∫1
4
[ (1+√ y )−( y−1 ) ]dy
A1 (RII )=∫0
1
[− y+√ y+2 ] dy
A1 (RII )=[− y22 + 23y32+2 x ]41
A1 (RII )=163
−136
A1 (RII )≈ 196u2
(2, 1)
A1R(II)
(0, 1)
(3, 4)
(0, 1)
Y ejes coordenados
Cálculo integral
A (RII )=A1 (RII )+A2 (RII )≈ 92u2
EJERCICIO 10 REGIÓN TIPO I
f(x) g(x)y=x y2=8−x
y1=√8−xy2=−√8−x
A1 (RI )=∫C
d
[ f ( x )−g1(x) ]dx
A1 (RI )= ∫2.37228
8
[ x−√8− x ]dx
A1 (RI )=[ x22 +(8−x )
32
32
]2.37228
8
A1 (RI )= (32 )−¿)
A1 (RI )≈20.2857U 2
A1 R(I)(2.37228, 2.37228)
(8, 0)
(8, 8)
Cálculo integral
A2 (RI )=∫a
b
[ f ( x )−g2(x )]dx
A2 (RI )= ∫−3.37228
8
[ x+√8−x ] dx
A2 (RI )=[ x22 −2 (8−x )
32
3 ]−3.37228
8
A2 (RI )=32−(−19.8808)
A2 (RI )≈51.8808u2
A (RI )=A2 (RI )−A1 (R I )≈31.5950u2
(8, 8)
(8, 0)
(-3.37228, 3.37228)
A1 R(I)
Cálculo integral
EJERCICIO 10 REGIÓN TIPO II
f(x) g(x)y=x y2=8−x
x=8− y2
A (RII )=∫c
d
[ g ( y )− f ( y ) ]dy
A (RII )= ∫−3.37228
2.37228
[(8− y2)− y ]dy
A (RII )=[8 y− y22 − y3
3 ] 2.37228−3.37228
A (RII )=11.7142−(−19.8808)
A (RII )≈31.5950u2
(2.37228, 2.37228)
(-3.37228,-3.37228)