Step-by-step Solution

1. Define the integral S₁

The problem defines the integral $S_i$ as $S_i = \int\limits_{\pi /8}^{3\pi /8} {f(x) \cdot {g_i}(x)dx}$. We need to find the value of $\frac{16S_1}{\pi}$. First, let's set up the integral for $S_1$ using the given functions:

$f(x) = \sin^2(x)$ $g_1(x) = 1$

So, the integral for $S_1$ is: $S_1 = \int\limits_{\pi /8}^{3\pi /8} {\sin^2(x) \cdot 1 \,dx} = \int\limits_{\pi /8}^{3\pi /8} {\sin^2(x)dx}$

2. Apply the King's Property of Definite Integrals

We will use the property $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$. In our case, $a = \frac{\pi}{8}$ and $b = \frac{3\pi}{8}$. So, $a+b = \frac{\pi}{8} + \frac{3\pi}{8} = \frac{4\pi}{8} = \frac{\pi}{2}$.

Applying this property to $S_1$: $S_1 = \int\limits_{\pi /8}^{3\pi /8} {\sin^2\left(\frac{\pi}{2} - x\right)dx}$ Since we know that $\sin(\frac{\pi}{2} - x) = \cos(x)$, we have $\sin^2(\frac{\pi}{2} - x) = \cos^2(x)$. Therefore, we get another expression for $S_1$: $S_1 = \int\limits_{\pi /8}^{3\pi /8} {\cos^2(x)dx}$

3. Combine the two expressions for S₁

Let's add the two expressions for $S_1$ that we have: $S_1 + S_1 = \int\limits_{\pi /8}^{3\pi /8} {\sin^2(x)dx} + \int\limits_{\pi /8}^{3\pi /8} {\cos^2(x)dx}$ $2S_1 = \int\limits_{\pi /8}^{3\pi /8} {(\sin^2(x) + \cos^2(x))dx}$ Using the trigonometric identity $\sin^2(x) + \cos^2(x) = 1$, the integral simplifies to: $2S_1 = \int\limits_{\pi /8}^{3\pi /8} {1 \,dx}$

4. Evaluate the simplified integral

Now, we evaluate the simple integral: $2S_1 = [x]_{\pi/8}^{3\pi/8}$ $2S_1 = \frac{3\pi}{8} - \frac{\pi}{8}$ $2S_1 = \frac{2\pi}{8}$ $2S_1 = \frac{\pi}{4}$

5. Solve for S₁

From the above result, we can find the value of $S_1$: $S_1 = \frac{\pi}{8}$

6. Calculate the final required value

The question asks for the value of $\frac{16S_1}{\pi}$. Substituting the value of $S_1$ we found: $\frac{16S_1}{\pi} = \frac{16 \left( \frac{\pi}{8} \right)}{\pi}$ $\frac{16S_1}{\pi} = \frac{16\pi}{8\pi}$ $\frac{16S_1}{\pi} = 2$

The information about $g_2(x)$ and $S_2$ is not needed to solve this problem.

Final Answer

The value of $\frac{16S_1}{\pi}$ is 2.