The formula to calculate the numerical approximation of a definite integral using Simpson's 3/8 Rule is:
\[ I = \frac{3h}{8} \left[ f(a) + 3f(a + h) + 3f(a + 2h) + f(b) \right] \]
Where:
Simpson's 3/8 Rule is a numerical integration technique for approximating definite integrals. It is a more accurate method than the standard Simpson's Rule as it uses cubic interpolation instead of quadratic, fitting a polynomial of degree three through four points instead of a polynomial of degree two through three points.
Let's assume the following values:
Step 1: Calculate the width of the subintervals:
\[ h = \frac{b - a}{n} = \frac{\frac{\pi}{2} - 0}{3} = \frac{\pi}{6} \]
Step 2: Calculate the integral:
\[ I = \frac{3h}{8} \left[ f(0) + 3f\left(\frac{\pi}{6}\right) + 3f\left(\frac{\pi}{3}\right) + f\left(\frac{\pi}{2}\right) \right] \]
Using the example function \( f(x) = \sin(x) \):
\[ I = \frac{3 \cdot \frac{\pi}{6}}{8} \left[ \sin(0) + 3\sin\left(\frac{\pi}{6}\right) + 3\sin\left(\frac{\pi}{3}\right) + \sin\left(\frac{\pi}{2}\right) \right] \]
\[ I = \frac{\pi}{16} \left[ 0 + 3 \cdot \frac{1}{2} + 3 \cdot \frac{\sqrt{3}}{2} + 1 \right] \]
\[ I = \frac{\pi}{16} \left[ 0 + \frac{3}{2} + \frac{3\sqrt{3}}{2} + 1 \right] = \frac{\pi}{16} \left( \frac{5 + 3\sqrt{3}}{2} \right) = \frac{\pi \left( 5 + 3\sqrt{3} \right)}{32} \]