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f @ x_ D := Sum @ a @ k D * x ^ k, 8 k, 0, n <D f @ x D

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(1)

Quit @D

H** Esercizio 4.3.9 ** L

n = 3;

f @ x_ D := Sum @ a @ k D * x ^ k, 8 k, 0, n <D f @ x D

a @ 0 D + x a @ 1 D + x

2

a @ 2 D + x

3

a @ 3 D

integrale = Integrate @ f @ x D , 8 x, -1, 1 <D quadratura = f B- 3 ’ 3 F + f B 3 ’ 3 F integrale - quadratura

2 a @ 0 D + 2 a @ 2 D 3

2 a @ 0 D + 2 a @ 2 D 3 0

H* Era necessario ? * L

f @ x_ D := x ^ 4 f @ x D

integrale = Integrate @ f @ x D , 8 x, -1, 1 <D quadratura = f B- 3 ’ 3 F + f B 3 ’ 3 F integrale - quadratura

x

4

2 5 2 9 8 45

H** Esercizio 4.3.10 ** L

n = 3;

f @ x_ D := Sum @ a @ k D * x ^ k, 8 k, 0, n <D f @ x D

a @ 0 D + x a @ 1 D + x

2

a @ 2 D + x

3

a @ 3 D

(2)

h = H b - a L  3;

x @ 0 D = a;

x @ 1 D = a + h;

x @ 2 D = b;

integrale = Integrate @ f @ x D , 8 x, a, b <D quadratura = 3 * h  4 * H 3 * f @ x @ 1 DD + f @ x @ 2 DDL integrale - quadratura

-a a @ 0 D + b a @ 0 D - 1 2

a

2

a @ 1 D + 1 2

b

2

a @ 1 D - 1 3

a

3

a @ 2 D + 1 3

b

3

a @ 2 D - 1 4

a

4

a @ 3 D + 1 4

b

4

a @ 3 D

1

4 H- a + b L a @ 0 D + b a @ 1 D + b

2

a @ 2 D + b

3

a @ 3 D + 3 a @ 0 D + a +

1

3 H- a + b L a @ 1 D + a + 1

3 H- a + b L

2

a @ 2 D + a + 1

3 H- a + b L

3

a @ 3 D

-a a @ 0 D + b a @ 0 D - 1 2

a

2

a @ 1 D + 1 2

b

2

a @ 1 D - 1 3

a

3

a @ 2 D + 1 3

b

3

a @ 2 D -

1 4

a

4

a @ 3 D + 1 4

b

4

a @ 3 D - 1

4 H- a + b L a @ 0 D + b a @ 1 D + b

2

a @ 2 D + b

3

a @ 3 D + 3 a @ 0 D + a +

1

3 H- a + b L a @ 1 D + a + 1

3 H- a + b L

2

a @ 2 D + a + 1

3 H- a + b L

3

a @ 3 D Simplify @ integrale - quadratura D

- 1

36 H a - b L

4

a @ 3 D

H** Esercizio 4.3.11 ** L

Clear @ f0, f1, f2, b0, b1, b2 D ; f0 @ x_ D := 1;

b0 = Integrate @ f0 @ x D , 8 x, -1, 1 <D f1 @ x_ D := x;

b1 = Integrate @ f1 @ x D , 8 x, -1, 1 <D f2 @ x_ D := x ^ 2;

b2 = Integrate @ f2 @ x D , 8 x, -1, 1 <D 2

0

2 3

Clear @ eq0, eq1, eq2, c0, c1, c2 D ;

eq0 = c0 * f0 @- 1 D + c1 * f0 @ 0 D + c2 * f0 @ 1 D == b0 eq1 = c0 * f1 @- 1 D + c1 * f1 @ 0 D + c2 * f1 @ 1 D == b1 eq2 = c0 * f2 @- 1 D + c1 * f2 @ 0 D + c2 * f2 @ 1 D == b2

c0 + c1 + c2 Š 2 -c0 + c2 Š 0

c0 + c2 Š 2 3

Solve @8 eq0, eq1, eq2 < , 8 c0, c1, c2 <D

:: c0 ® 1 3

, c1 ® 4 3

, c2 ® 1 3 >>

H** Esercizio 4.3.12 ** L

2 Integrazione4.nb

(3)

Clear @ f0, f1, f2, b0, b1, b2 D ; f0 @ x_ D := 1;

b0 = Integrate @ f0 @ x D , 8 x, 0, 2 <D f1 @ x_ D := x;

b1 = Integrate @ f1 @ x D , 8 x, 0, 2 <D f2 @ x_ D := x ^ 2;

b2 = Integrate @ f2 @ x D , 8 x, 0, 2 <D 2

2

8 3

Clear @ eq0, eq1, eq2, c0, c1, c2 D ;

eq0 = c0 * f0 @ 0 D + c1 * f0 @ 1 D + c2 * f0 @ 2 D == b0 eq1 = c0 * f1 @ 0 D + c1 * f1 @ 1 D + c2 * f1 @ 2 D == b1 eq2 = c0 * f2 @ 0 D + c1 * f2 @ 1 D + c2 * f2 @ 2 D == b2 c0 + c1 + c2 Š 2

c1 + 2 c2 Š 2

c1 + 4 c2 Š 8 3

Solve @8 eq0, eq1, eq2 < , 8 c0, c1, c2 <D

:: c0 ® 1 3

, c1 ® 4 3

, c2 ® 1 3 >>

H** Esercizio 4.3.13 ** L

f0 @ x_ D := 1;

b0 = Integrate @ f0 @ x D , 8 x, 0, 1 <D f1 @ x_ D := x

b1 = Integrate @ f1 @ x D , 8 x, 0, 1 <D f2 @ x_ D := x ^ 2

b2 = Integrate @ f2 @ x D , 8 x, 0, 1 <D f3 @ x_ D := x ^ 3

b3 = Integrate @ f3 @ x D , 8 x, 0, 1 <D 1

1 2 1 3 1 4

eq0 = c0 * f0 @ 0 D + c1 * f0 @ x1 D Š b0 eq1 = c0 * f1 @ 0 D + c1 * f1 @ x1 D Š b1 eq2 = c0 * f2 @ 0 D + c1 * f2 @ x1 D Š b2 c0 + c1 Š 1

c1 x1 Š 1 2

c1 x1

2

Š 1 3

Integrazione4.nb 3

(4)

sol = Solve @8 eq0, eq1, eq2 < , 8 c0, c1, x1 <D@@ 1 DD

: c0 ® 1 4

, c1 ® 3 4

, x1 ® 2 3 >

H c0 * f3 @ 0 D + c1 * f3 @ x1 DL  . sol 2

9

4 Integrazione4.nb

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Plot @ f @ x D , 8 x, 0, 1 < , PlotStyle ® 8 RGBColor @ 0, 0, 1 D , Thickness @ 0.01 D

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