Integrales Region Tipo 2

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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)

Integrales Region Tipo 2

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