## Procedimientos numéricos

### Aro

$\begin{array}{l}\frac{{d}^{2}x}{d{t}^{2}}=-\frac{\mu g}{2\pi }\underset{0}{\overset{2\pi }{\int }}\frac{v-\omega R\text{sin}\theta }{\sqrt{{v}^{2}+{\omega }^{2}{R}^{2}-2v\omega R\text{sin}\theta }}d\theta \\ \frac{{d}^{2}\varphi }{d{t}^{2}}=-\frac{\mu g}{2\pi {R}^{2}}\underset{0}{\overset{2\pi }{\int }}\left(\frac{R\left(\omega R-v\text{sin}\theta \right)}{\sqrt{{v}^{2}+{\omega }^{2}{R}^{2}-2v\omega R\text{sin}\theta }}\right)d\theta \end{array}$

### Disco

$\begin{array}{l}\frac{{d}^{2}x}{d{t}^{2}}=-\frac{\mu g}{\pi {R}^{2}}\underset{0}{\overset{2\pi }{\int }}d\theta \underset{0}{\overset{R}{\int }}\frac{v-\omega \text{\hspace{0.17em}}r\text{sin}\theta }{\sqrt{{v}^{2}+{\omega }^{2}{r}^{2}-2v\omega \text{\hspace{0.17em}}r\text{sin}\theta }}rdr\\ \frac{{d}^{2}\varphi }{d{t}^{2}}=-\frac{2\mu g}{\pi {R}^{4}}\underset{0}{\overset{2\pi }{\int }}d\theta \underset{0}{\overset{R}{\int }}\left(\frac{{r}^{2}\left(\omega \text{\hspace{0.17em}}r-v\text{sin}\theta \right)}{\sqrt{{v}^{2}+{\omega }^{2}{r}^{2}-2v\omega \text{\hspace{0.17em}}r\text{sin}\theta }}\right)dr\end{array}$

 `public abstract class Simpson { int n; final double a=0.0; final double b=2*Math.PI; public Simpson(int n){ if(n%2==1) n++; //hace que el número de divisiones sea par this.n=n; } public double integral(){ double h=(b-a)/n; double suma=f(a)+f(b); for(int i=1; i