Evaluate the following integration using Romberg integration. | Numerical Method 2078

Numerical Method20785 marksNumerical: Romberg integrationRomberg integration

Evaluate the following integration using Romberg integration. $\int_{0}^{1} \frac{\sin^2 x}{x} dx$ [5]

Evaluate the following integration using Romberg integration. $$\int_{0}^{1} \frac{\sin^2 x}{x} dx$$ [5]

Integrand: $f(x) = \dfrac{\sin^2 x}{x}$ - Limits: $a = 0$ , $b = 1$ - Interval length: $b - a = 1$ Removable singularity at $x=0$ : $\lim{x\to 0}\frac{\sin^2 x}{x} = \lim{x\to0}\frac{x^2}{x} = 0$ , so $f(0)=0$ . $x$ $\sin x$ $\sin^2 x$ $f(x)=\sin^2 x/x$ ------------ $0$ $0$ $0$ $0$ $0.25$

Integrand: $f(x) = \dfrac{\sin^2 x}{x}$ - Limits: $a = 0$, $b = 1$ - Interval length: $b - a = 1$ Removable singularity at $x=0$: $\lim{x\to 0}\frac{\sin^2 x}{x} = \lim{x\to0}\frac{x^2}{x} = 0$, so $f(0)=0$. $x$ $\sin x$ $\sin^2 x$ $f(x)=\sin^2 x/x$ ------------ $0$ $0$ $0$ $0$ $0.25$