Calcular el centro geométrico de las funciones 3x+4y=24, y=0, x=0

y=24−3 x4

x=∫a

b

x (f (x ))dx

∫a

b

f ( x )dx

f ( x )=∫a

b

( 24−3x4

)dx

Area=14∫0

8

(24−3 x)dx

A=14 [24 x−3 x22 ]

0

8

A=14 [−24 (0 )+ 3 (0 )2

2+24 (8 )−3 (8 )2

2 ]A=24u2

(x ) f ( x )=∫a

b

x ( 24−3 x4

)dx

¿ 14∫0

8

x (24−3 x)dx

¿ 14∫0

8

(24 x−3 x2 )dx

¿14 [12 x2−3 x33 ]

0

8

¿ 14

[12 (8 )2−(8 )3 ]

¿ 14(256)

¿64

x=∫a

b

x (f (x ))dx

∫a

b

f ( x )dx

x=6424

=83

A . y=12∫a

b

( f ( x ) )2dx

A . y=12∫a

b

( 24−3 x4

)2

dx

A . y= 132

∫0

8

(576−144 x+9 x2)

A . y=132 [576 x−144 x

2

2+9 x3

3 ]0

8

A . y=132 [576 x−144 x

2

2+9 x3

3 ]0

8

A . y= 132 [576 (8)−144 (8)2

2+9 (8)3

3 ]A . y= 1

32(1536)

A . y=48

y= 4824

=2

c=( 83,2)

Calcular el centro geométrico de las funciones y=3 x2−x3 , y=3 x−x2

P.Int.

3 x2−x3=3 x−x2x3−x2−3 x2+3x=0

x3−4 x2+3 x=0x ( x2−4 x+3 )=0x (x−3)(x−1)=0x1=0 P1(0,0)

x2=1P2(1,2) x3=3 P1(3,0)

De 0 a 1:

x=∫a

b

x ( f ( x )−g (x ) )dx

∫a

b

( f ( x )−g ( x ) )dx

A=∫0

1

( ( fx )−g ( x ) )dxA=∫0

1

(3 x−x2−3 x2+x3 )dx A=∫0

1

(x3−4 x2+3 x )dx

A=[ x 44 −4 x3

3+3 x2

2 ]0

1

A=14−43+ 32= 512u2

A . x=∫0

1

x ( f ( x )−g(x))dx

A . x=∫0

1

x (3 x−x2−3 x2+x3 )dx

A . x=∫0

1

(x4−4 x3+3 x2 )dx A . x=[ x55 −4 x4

4+3 x3

3 ]0

1

A . x=15−1+1x=1

5 (125 )=0.48

A . y=12∫a

b

[ ( f ( x ))2−(g ( x ))2 ]dx

A . y=12∫0

1

[ (3 x−x2 )2−(3 x2−x3 )2 ]dx A . y=12∫0

1

[9 x2−6 x3+ x4−9 x4+6 x5+x6 ] dx

A . y=12∫0

1

[9 x2−6 x3−8 x4+6 x5−x6 ] dxA . y=12 [ 9 x33 −

6 x4

4−8 x5

5+6 x6

6−x7

7 ]0

1

A . y=12 [3−32−85 +1−1

7 ]y= 53140 ( 125 )=0.9

C=(0.48 ,0.9)

De 1 a 3:

x=∫a

b

x (g ( x )−f (x ) )dx

∫a

b

(g ( x )−f ( x ) )dx

A=∫a

b

(g (x )−f ( x ) )dx

A=∫1

3

(−x3+4 x2−3 x )dx

A=[−x44 +4 x3

3−3 x2

2 ]1

3

A=[−(3 )4

4+4 (3 )3

3−3 (3 )2

2− x

4

4−4 x

3

3+3 x

2

2 ]1

3

A=83u2

A . x=∫1

3

x (g ( x )−f (x))dx

A . x=∫1

3

(−x4+4 x3−3 x2)dx

A . x=[−x55 +4 x4

4−3 x3

3 ]1

3

A . x=−35

5+4 (3 )4

4−3 (3 )3

3+ 15−1+1

A . x=285

x=285 ( 38 )=2110