Esame di Stato 2001 PNI – 6

Con uno dei metodi di quadratura studiati, si calcoli un’approssimazione dell’integrale definito \displaystyle \int_{0}^\pi\, \sin{x}\,dx e si confronti il risultato con il valore esatto dell’integrale.

Valore esatto

\displaystyle \int_{0}^\pi\ \sin{x}\ dx=\big[-\cos x\big]_0^\pi=(-\cos{\pi})-(-\cos\ {0})=1+1=2

Integrazione numerica

Approssima l’integrale con tutti i metodi trattati in classe

  • Metodo del punto centrale, x = (a+b)/2
  • Metodo dei rettangoli, altezze di sinistra, x = a, a+h, a+2h, …, a+(n-1)h
  • Metodo dei rettangoli, altezze al centro, x = a+h/2, a+h+h/2, a+2h+h/2, …, a+(n-1)h+h/2
  • Metodo dei rettangoli, altezze di destra, x = a+h, a+2h, …, a+(n-1)h, b
  • Metodo dei trapezi, x = a, a+h, a+2h, …, a+(n-1)h, b
  • Metodo delle parabole, x = a, a+h, a+2h, …, a+(n-1)h, b

Metodo del punto centrale

  • \displaystyle m = \frac{a+b}{2}= \frac{0+\pi}{2} = \frac{\pi}{2}
  • \displaystyle f(m) = f\left(\frac{\pi}{2}\right) = \sin{\frac{\pi}{2}} = 1
  • Integrale definito approssimato = \displaystyle (b - a)\cdot f(m) = (\pi -0)\cdot 1 = \pi

Metodo dei rettangoli

Altezze di sinistra

n h x_i y_i \displaystyle h (y_0)

1

 = \displaystyle \frac{\pi-0}{1}

= \pi

 0 0 = \pi\cdot 0

= 0
n h x_i y_i \displaystyle h (y_0+y_1)

2

 = \displaystyle \frac{\pi-0}{2}

= \displaystyle \frac{\pi}{2}

 0

\displaystyle \frac{\pi}{2}

 0

1

 = \displaystyle\frac{\pi}{2}(0+1)

= \displaystyle \frac{1}{2}\,\pi

= 1.570796
n h x_i y_i \displaystyle h (y_0+y_1+y_2)

3

 = \displaystyle \frac{\pi-0}{3}

= \displaystyle \frac{\pi}{3}

 0

\displaystyle\frac{\pi}{3}

\displaystyle\frac{2}{3}\,\pi

 0

\displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{\sqrt{3}}{2}

 = \displaystyle\frac{\pi}{3}\left(0+\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}\right)

= \displaystyle\frac{\sqrt{3}}{3}\,\pi

= 1.813799
n h x_i y_i \displaystyle h (y_0+y_1+y_2+y_3)

4

 = \displaystyle \frac{\pi-0}{4}

= \displaystyle \frac{\pi}{4}

 0

\displaystyle\frac{\pi}{4}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{3}{4}\,\pi

 0

\displaystyle\frac{\sqrt{2}}{2}

1

\displaystyle\frac{\sqrt{2}}{2}

 = \displaystyle\frac{\pi}{4}\left(0+\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}\right)

= \displaystyle\frac{\sqrt{2}+1}{4}\,\pi

= 1.896119
n h x_i y_i \displaystyle h (y_0+y_1+y_2+y_3+y_4+y_5)

6

 = \displaystyle \frac{\pi-0}{6}

= \displaystyle \frac{\pi}{6}

 0

\displaystyle\frac{\pi}{6}

\displaystyle\frac{\pi}{3}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{2}{3}\,\pi

\displaystyle\frac{5}{6}\,\pi

 0

\displaystyle\frac{1}{2}

\displaystyle\frac{\sqrt{3}}{2}

1

\displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{1}{2}

 = \displaystyle\frac{\pi}{6}\left(0+\frac{1}{2}+\frac{\sqrt{3}}{2}+1+\frac{\sqrt{3}}{2}+\frac{1}{2}\right)

= \displaystyle\frac{\sqrt{3}+2}{6}\,\pi

= 1.954097

Altezze di destra

n h x_i y_i \displaystyle h (y_1)

1

 = \displaystyle \frac{\pi-0}{1}

= \pi

 \pi 0 = \pi\cdot 0

= 0
n h x_i y_i \displaystyle h (y_1+y_2)

2

 = \displaystyle \frac{\pi-0}{2}

= \displaystyle \frac{\pi}{2}

 \displaystyle \frac{\pi}{2}

\pi

 1

0

 = \displaystyle\frac{\pi}{2}(1+0)

= \displaystyle \frac{1}{2}\,\pi

= 1.570796
n h x_i y_i \displaystyle h (y_1+y_2+y_3)

3

 = \displaystyle \frac{\pi-0}{3}

= \displaystyle \frac{\pi}{3}

 \displaystyle\frac{\pi}{3}

\displaystyle\frac{2}{3}\,\pi

\pi

 \displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{\sqrt{3}}{2}

0

 = \displaystyle\frac{\pi}{3}\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}+0\right)

= \displaystyle\frac{\sqrt{3}}{3}\,\pi

= 1.813799
n h x_i y_i \displaystyle h (y_1+y_2+y_3+y_4)

4

 = \displaystyle \frac{\pi-0}{4}

= \displaystyle \frac{\pi}{4}

 \displaystyle\frac{\pi}{4}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{3}{4}\,\pi

\pi

 \displaystyle\frac{\sqrt{2}}{2}

1

\displaystyle\frac{\sqrt{2}}{2}

0

 = \displaystyle\frac{\pi}{4}\left(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\right)

= \displaystyle\frac{\sqrt{2}+1}{4}\,\pi

= 1.896119
n h x_i y_i \displaystyle h (y_1+y_2+y_3+y_4+y_5+y_6)

6

 = \displaystyle \frac{\pi-0}{6}

= \displaystyle \frac{\pi}{6}

 \displaystyle\frac{\pi}{6}

\displaystyle\frac{\pi}{3}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{2}{3}\,\pi

\displaystyle\frac{5}{6}\,\pi

\pi

 \displaystyle\frac{1}{2}

\displaystyle\frac{\sqrt{3}}{2}

1

\displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{1}{2}

0

 = \displaystyle\frac{\pi}{6}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}+1+\frac{\sqrt{3}}{2}+\frac{1}{2}+0\right)

= \displaystyle\frac{\sqrt{3}+2}{6}\,\pi

= 1.954097

Altezze nel punto centrale

n h h/2 x_i^* y_i^* \displaystyle h (y_0^*)

1

 = \displaystyle \frac{\pi-0}{1}

= \pi

 = \displaystyle \pi \cdot \frac{1}{2}

= \displaystyle \frac{\pi}{2}

 \displaystyle \frac{\pi}{2} 1 = \pi\cdot 1

= \pi

= 3.14159…
n h h/2 x_i^* y_i^* \displaystyle h (y_0^*+y_1^*)

2

 = \displaystyle \frac{\pi-0}{2}

= \displaystyle \frac{\pi}{2}

 = \displaystyle \frac{\pi}{2}\cdot \frac{1}{2}

= \displaystyle \frac{\pi}{4}

 \displaystyle \frac{\pi}{4}

\displaystyle \frac{3}{4}\,\pi

 \displaystyle \frac{\sqrt{2}}{2}

\displaystyle \frac{\sqrt{2}}{2}

 = \displaystyle \frac{\pi}{2}\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)

= \displaystyle \frac{\sqrt{2}}{2}\,\pi

= 2.2214…
n h h/2 x_i^* y_i^* \displaystyle h (y_0^*+y_1^*+y_2^*)

3

 = \displaystyle \frac{\pi-0}{3}

= \displaystyle \frac{\pi}{3}

 = \displaystyle \frac{\pi}{3}\cdot \frac{1}{2}

= \displaystyle \frac{\pi}{6}

 \displaystyle\frac{\pi}{6}

\displaystyle\frac{\pi}{2}\,\pi

\displaystyle\frac{5}{6}\,\pi

 \displaystyle\frac{1}{2}

1

\displaystyle\frac{1}{2}

 = \displaystyle\frac{\pi}{3}\left(\frac{1}{2}+1+\frac{1}{2}\right)

= \displaystyle\frac{2}{3}\,\pi

= 2.0943…

Metodo dei trapezi

n h x_i y_i \displaystyle \frac{h}{2} (y_0+y_1)

1

 = \displaystyle \frac{\pi-0}{1}

= \pi

 o

\pi

 0

0

 = \displaystyle \frac{\pi}{2} (0+0)

= \displaystyle \frac{\pi}{2}\cdot 0

= 0
n h x_i y_i \displaystyle \frac{h}{2} (y_0+2y_1+y_2)

2

 = \displaystyle \frac{\pi-0}{2}

= \displaystyle \frac{\pi}{2}

 0

\displaystyle \frac{\pi}{2}

\pi

 0

1

0

 = \displaystyle\frac{\pi}{4}(0+2\cdot 1+0)

= \displaystyle \frac{1}{2}\,\pi

= 1.570796
n h x_i y_i \displaystyle \frac{h}{2} (y_0+2y_1+2y_2+y_3)

3

 = \displaystyle \frac{\pi-0}{3}

= \displaystyle \frac{\pi}{3}

 0

\displaystyle\frac{\pi}{3}

\displaystyle\frac{2}{3}\,\pi

\pi

 0

\displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{\sqrt{3}}{2}

0

 = \displaystyle\frac{\pi}{6}\left(0+2\cdot \frac{\sqrt{3}}{2}+2\cdot \frac{\sqrt{3}}{2}+0\right)

= \displaystyle\frac{\sqrt{3}}{3}\,\pi

= 1.813799
n h x_i y_i \displaystyle \frac{h}{2} (y_0+2 y_1+2 y_2+2 y_3+y_4)

4

 = \displaystyle \frac{\pi-0}{4}

= \displaystyle \frac{\pi}{4}

 0

\displaystyle\frac{\pi}{4}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{3}{4}\,\pi

\pi

 0

\displaystyle\frac{\sqrt{2}}{2}

1

\displaystyle\frac{\sqrt{2}}{2}

0

 = \displaystyle\frac{\pi}{8}\left(0+2\cdot\frac{\sqrt{2}}{2}+2\cdot 1+2\cdot\frac{\sqrt{2}}{2}+0\right)

= \displaystyle\frac{\sqrt{2}+1}{4}\,\pi

= 1.896119
n h x_i y_i \displaystyle \frac{h}{2} (y_0+2 y_1+2 y_2+2 y_3+2 y_4+2 y_5+y_6)

6

 = \displaystyle \frac{\pi-0}{6}

= \displaystyle \frac{\pi}{6}

 0

\displaystyle\frac{\pi}{6}

\displaystyle\frac{\pi}{3}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{2}{3}\,\pi

\displaystyle\frac{5}{6}\,\pi

\pi

 0

\displaystyle\frac{1}{2}

\displaystyle\frac{\sqrt{3}}{2}

1

\displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{1}{2}

0

 = \displaystyle\frac{\pi}{12}\left(0+2\cdot\frac{1}{2}+2\cdot\frac{\sqrt{3}}{2}+2\cdot 1+2\cdot\frac{\sqrt{3}}{2}+2\cdot\frac{1}{2}+0\right)

= \displaystyle\frac{\sqrt{3}+2}{6}\,\pi

= 1.954097

Metodo delle parabole

n h x_i y_i \displaystyle \frac{h}{3} (y_0+4 y_1+y_2)

2

 = \displaystyle \frac{\pi-0}{2}

= \displaystyle \frac{\pi}{2}

 0

\displaystyle \frac{\pi}{2}

\pi

 0

1

0

 = \displaystyle\frac{\pi}{6}(0+4\cdot 1+0)

= \displaystyle \frac{2}{3}\,\pi

= 2.094395
n h x_i y_i \displaystyle \frac{h}{3} (y_0+4 y_1+2 y_2+4 y_3+y_4)

4

 = \displaystyle \frac{\pi-0}{4}

= \displaystyle \frac{\pi}{4}

 0

\displaystyle\frac{\pi}{4}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{3}{4}\,\pi

\pi

 0

\displaystyle\frac{\sqrt{2}}{2}

1

\displaystyle\frac{\sqrt{2}}{2}

0

 = \displaystyle\frac{\pi}{12}\left(0+4\cdot\frac{\sqrt{2}}{2}+2\cdot 1+4\cdot\frac{\sqrt{2}}{2}+0\right)

= \displaystyle\frac{2\sqrt{2}+1}{6}\,\pi

= 2.004560
n h x_i y_i \displaystyle \frac{h}{3} (y_0+4 y_1+2 y_2+4 y_3+2 y_4+4 y_5 +y_6)

6

 = \displaystyle \frac{\pi-0}{6}

= \displaystyle \frac{\pi}{6}

 0

\displaystyle\frac{\pi}{6}

\displaystyle\frac{\pi}{3}

\displaystyle\frac{\pi}{2}

\displaystyle\frac{2}{3}\,\pi

\displaystyle\frac{5}{6}\,\pi

\pi

 0

\displaystyle\frac{1}{2}

\displaystyle\frac{\sqrt{3}}{2}

1

\displaystyle\frac{\sqrt{3}}{2}

\displaystyle\frac{1}{2}

0

 = \displaystyle\frac{\pi}{18}\left(0+4\cdot\frac{1}{2}+2\cdot\frac{\sqrt{3}}{2}+4\cdot 1+2\cdot\frac{\sqrt{3}}{2}+4\cdot\frac{1}{2}+0\right)

= \displaystyle\frac{\sqrt{3}+4}{9}\,\pi

= 2.000863

Riepilogo

n Punto
centrale		 Rettangoli (con scelta delle altezze) Trapezi Parabole
di sinistra al centro di destra
1 \pi 0 \pi 0 0
2 \displaystyle \frac{1}{2}\,\pi \displaystyle \frac{\sqrt{2}}{2}\,\pi \displaystyle \frac{1}{2}\,\pi \displaystyle \frac{1}{2}\,\pi \displaystyle \frac{2}{3}\,\pi
3 \displaystyle\frac{\sqrt{3}}{3}\,\pi \displaystyle\frac{2}{3}\,\pi \displaystyle\frac{\sqrt{3}}{3}\,\pi \displaystyle\frac{\sqrt{3}}{3}\,\pi
4 \displaystyle\frac{\sqrt{2}+1}{4}\,\pi \displaystyle\frac{\sqrt{2}+1}{4}\,\pi \displaystyle\frac{\sqrt{2}+1}{4}\,\pi \displaystyle\frac{2\sqrt{2}+1}{6}\,\pi
5
6 \displaystyle\frac{\sqrt{3}+2}{6}\,\pi \displaystyle\frac{\sqrt{3}+2}{6}\,\pi \displaystyle\frac{\sqrt{3}+2}{6}\,\pi \displaystyle\frac{\sqrt{3}+4}{9}\,\pi

Riepilogo numerico

n Punto
centrale		 Rettangoli (con scelta delle altezze) Trapezi Parabole
di sinistra al centro di destra
1 3.1415… 0 3.1415… 0 0
2 1.570796 2.2214… 1.570796 1.570796 2.094395
3 1.813799 2.0943… 1.813799 1.813799
4 1.896119 1.896119 1.896119 2.004560
5
6 1.954097 1.954097 1.954097 2.000863