====== Integrali notevoli con rotazione del sistema di riferimento ======

===== Momenti statici =====

Le formule per il calcolo dei momenti statici nel caso di rotazione del sistema di riferimento di un angolo $\theta$ sono

$$S_{z}^{\odot} = S_z \cdot \cos \theta + S_y \cdot \sin \theta$$

$$S_{y}^{\odot} = - S_z \cdot \sin \theta + S_y \cdot \cos \theta $$

Per il calcolo analitico vedi la pagina del wiki sulla [[scienza_costruzioni:geometria_delle_aree|geometria delle aree]].

===== Momenti di inerzia =====

Le formule per il calcolo dei momenti di inerzia nel caso di rotazione del sistema di riferimento di un angolo $\theta$ sono

$$I_{yy}^\odot = I_{yy} \cos ^2 \theta - 2 I_{yz} \sin \theta \cos \theta + I_{zz} \sin ^2 \theta$$

$$I_{zz}^\odot = I_{yy} \sin ^2 \theta + 2 I_{yz} \sin \theta \cos \theta + I_{zz} \cos ^2 \theta$$

$$I_{yz}^\odot = \left( I_{yy} - I_{zz} \right) \sin \theta \cos \theta + I_{yz} \left( \cos^2 \theta - \sin^2 \theta \right) = \frac{I_{yy} - I_{zz}}{2} \sin 2 \theta + I_{yz} \cos 2 \theta$$


Per maggiori dettagli su come si determinano tali formule si rimanda alla sezione del wiki sulla [[scienza_costruzioni:geometria_delle_aree|geometria delle aree]].

===== Momenti di terzo ordine =====

Le formule che ci permettono di calcolare i momenti di terzo ordine in caro di rotazione di angolo $\theta$ sono

$$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \\
\cos^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \cos^2 \theta \, \sin \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 3 \cos \theta \, \sin^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \\
- \sin^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 3 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \\
- \sin \theta \, \cos^2 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - \sin^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\
+ \cos^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^2 \theta \, \cos \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \\
\sin^2 \theta \, \cos \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\
+ \sin^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin \theta \, \cos^2 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

Si riporta di seguito il dettaglio dei calcoli

$$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\
\iint \limits_{S} \left( \cos^3 \theta \, y^3 + 3 \cos^2 \theta \, \sin \theta \, y^2 \, z + 3 \cos \theta \, \sin^2 \theta \, y \, z^2 + \sin^3 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\
\cos^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \cos^2 \theta \, \sin \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 3 \cos \theta \, \sin^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( - \sin \theta \, y + \cos \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\
\iint \limits_{S} \left( - \sin^3 \theta \, y^3 + 3 \sin^2 \theta \, \cos \theta \, y^2 \, z - 3 \sin \theta \, \cos^2 \theta \, y \, z^2 + \cos^3 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\
- \sin^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 3 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^2 \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\
\iint \limits_{S} \left( \cos^2 \theta \, y^2 + 2 \sin \theta \, \cos \theta \, y \, z + \sin^2 \theta \, z^2 \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\ 
\iint \limits_{S} \left( - \sin \theta \, \cos^2 \theta \, y^3 - 2 \sin^2 \theta \, \cos \theta \, y^2 \, z - \sin^3 \theta \, y \, z^2 + \cos^3 \theta \, y^2 \, z + 2 \sin \theta \, \cos^2 \theta \, y \, z^2 + \sin^2 \theta \, \cos \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\
- \sin \theta \, \cos^2 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - \sin^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\
+ \cos^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^2 \theta \, \cos \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right)^2 \; \mathrm{d}y \mathrm{d}z = \\
\iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( \sin^2 \theta \, y^2 - 2 \sin \theta \, \cos \theta \, y \, z + \cos^2 \theta \, z^2 \right) \; \mathrm{d}y \mathrm{d}z = \\
\iint \limits_{S} \left( \sin^2 \theta \, \cos \theta \, y^3 - 2 \sin \theta \, \cos^2 \theta \, y^2 \, z + \cos^3 \theta \, y \, z^2 + \sin^3 \theta \, y^2 \, z - 2 \sin^2 \theta \, \cos \theta \, y \, z^2 + \sin \theta \, \cos^2 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\
\sin^2 \theta \, \cos \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\
+ \sin^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin \theta \, \cos^2 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$