Let's complete the calculations for \( A_1 \) and \( A_2 \) to find the total area.

### Step 1: Calculate \( A_1 \)
\[
A_1 = \frac{1535}{12288} + \frac{2}{3}
\]

To simplify:
\[
A_1 = \frac{1535}{12288} + \frac{2}{3} \times \frac{4096}{4096} = \frac{1535}{12288} + \frac{8192}{12288} = \frac{9727}{12288}
\]

Simplify the fraction:
\[
A_1 \approx 0.791
\]

### Step 2: Calculate \( A_2 \)
\[
A_2 = 256 - \frac{512}{3} - \left(\frac{96}{1536} - \frac{1}{1536}\right)
\]
Calculate each term:

\[
256 - \frac{512}{3} = 256 - 170.67 = 85.33
\]
\[
\frac{96}{1536} = \frac{1}{16} = 0.0625
\]
\[
\frac{95}{1536} = \frac{96 - 1}{1536} = 0.0618
\]

So,
\[
A_2 = 85.33 - 0.0618 \approx 85.27
\]

### Step 3: Calculate the total area
Add \( A_1 \) and \( A_2 \):
\[
\text{Total Area} = A_1 + A_2 \approx 0.791 + 85.27 = 86.061
\]

Thus, the area of the region enclosed by the curves is approximately \( \boxed{86.061} \) square units.

Would you like further clarification on any part of the process, or have any more questions? Here are 5 related questions:

1. How do you handle integrals with absolute values in similar problems?
2. What happens if the curves enclose multiple regions? How do you find the total area?
3. How would you approach this problem if the functions involved trigonometric equations?
4. Can you find the volume of a solid of revolution generated by rotating this region about the x-axis?
5. How would the problem change if the curves intersect at complex points?

**Tip:** When dealing with multiple integrals, carefully check the intervals and the functions on those intervals to avoid mistakes in the calculation.

Learn how to calculate the total area enclosed by curves using integration and numerical approximation. Step-by-step guide with detailed calculations and explanations. Suitable for advanced high school and college students.

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