If the area bounded by circle `x ^(2) + y^(2)=4,` the parabola `y = x ^(2) + x+1` and the curve `y = [sin ^(2) ""(x)/(4) +cos ""(x)/(4)], ` (where [] denotes the greats integer function) and x-axis is `(sqrt3 + (2pi)/(3) - (1)/(k)),` then the numerical quantitity is should be :

Answer

To solve the problem, we need to find the area bounded by the circle \(x^2 + y^2 = 4\), the parabola \(y = x^2 + x + 1\), and the curve \(y = \lfloor \sin^2\left(\frac{x}{4}\right) + \cos\left(\frac{x}{4}\right) \rfloor\) along with the x-axis. The area is given as \( \sqrt{3} + \frac{2\pi}{3} - \frac{1}{k} \). We need to find the numerical value of \(k\).

### Step-by-Step Solution:

1. **Identify the curves:**
- The circle \(x^2 + y^2 = 4\) has a center at \((0, 0)\) and a radius of \(2\).
- The parabola \(y = x^2 + x + 1\) can be rewritten in vertex form as \(y = (x + \frac{1}{2})^2 + \frac{3}{4}\).
- The function \(y = \lfloor \sin^2\left(\frac{x}{4}\right) + \cos\left(\frac{x}{4}\right) \rfloor\) varies based on the values of \(\sin\) and \(\cos\).

2. **Graph the functions:**
- Plot the circle, parabola, and the constant function derived from the greatest integer function. The constant function will be \(y = 1\) over the relevant interval.

3. **Determine intersection points:**
- Find the points where the parabola intersects the circle and the x-axis.
- Solve \(x^2 + x + 1 = 2\) to find the intersection with the circle.
- Solve \(x^2 + x + 1 = 0\) for the x-axis.

4. **Calculate the area:**
- Break the area into segments based on the intersection points.
- For each segment, set up the integral to find the area between the curves and the x-axis.

5. **Evaluate the integrals:**
- For the area under the circle from \(x = -2\) to \(x = -\sqrt{3}\):
\[ A_1 = \int_{-2}^{-\sqrt{3}} \sqrt{4 - x^2} \, dx \]
- For the area under the parabola from \(x = -\sqrt{3}\) to \(x = -1\):
\[ A_2 = \int_{-\sqrt{3}}^{-1} (x^2 + x + 1) \, dx \]
- For the area under the constant function from \(x = -1\) to \(x = 0\):
\[ A_3 = \int_{-1}^{0} 1 \, dx \]
- For the area under the constant function from \(x = 0\) to \(x = \sqrt{3}\):
\[ A_4 = \int_{0}^{\sqrt{3}} 1 \, dx \]
- For the area under the circle from \(x = \sqrt{3}\) to \(x = 2\):
\[ A_5 = \int_{\sqrt{3}}^{2} \sqrt{4 - x^2} \, dx \]

6. **Combine the areas:**
- The total area \(A\) is given by:
\[ A = A_1 + A_2 + A_3 + A_4 + A_5 \]

7. **Set the total area equal to the given expression:**
- Set \(A\) equal to \(\sqrt{3} + \frac{2\pi}{3} - \frac{1}{k}\) and solve for \(k\).

8. **Final calculation:**
- After evaluating the integrals and simplifying, compare the results to find \(k\).