Given the integral \( A = \int_{0}^{32} \left( \frac{1}{2} - 2\sqrt{2x} \right) dx \), let's solve it with the new limits.

### Step 1: Break the integral into parts
\[
A = \int_{0}^{32} \frac{1}{2} \, dx - \int_{0}^{32} 2\sqrt{2x} \, dx
\]

### Step 2: Integrate each term separately

#### First integral:
\[
\int_{0}^{32} \frac{1}{2} \, dx = \frac{1}{2} \int_{0}^{32} dx = \frac{1}{2} \left[ x \right]_{0}^{32} = \frac{1}{2} \times 32 = 16
\]

#### Second integral:
For the integral \(\int 2\sqrt{2x} \, dx\), we perform a substitution.

Let \( u = 2x \), then \( du = 2 \, dx \), so \( dx = \frac{du}{2} \). The integral becomes:
\[
\int 2\sqrt{2x} \, dx = \int 2\sqrt{u} \cdot \frac{du}{2} = \int \sqrt{u} \, du
\]
Now, integrate \( \sqrt{u} \) using the power rule:
\[
\int \sqrt{u} \, du = \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} u^{\frac{3}{2}}
\]
Substitute back \( u = 2x \):
\[
\frac{2}{3} (2x)^{\frac{3}{2}}
\]
Now, evaluate this from \( x = 0 \) to \( x = 32 \):
\[
\left[\frac{2}{3} (2x)^{\frac{3}{2}}\right]_{0}^{32} = \frac{2}{3} \left[(2 \cdot 32)^{\frac{3}{2}} - (2 \cdot 0)^{\frac{3}{2}}\right] = \frac{2}{3} \times 64^{\frac{3}{2}} = \frac{2}{3} \times 512 \times 8 = \frac{8192}{3}
\]

### Step 3: Combine results
\[
A = 16 - \frac{8192}{3} = \frac{48}{3} - \frac{8192}{3} = \frac{48 - 8192}{3} = \frac{-8144}{3}
\]

Thus, the value of \( A \) is \( \boxed{-2714.67} \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the significance of a negative result in a definite integral?
2. How can you interpret this integral graphically?
3. What does the result of this integral tell us about the function being integrated?
4. How do different limits of integration affect the value of the definite integral?
5. How do you deal with square roots in integration?
6. What are some common mistakes to avoid when solving definite integrals?
7. What other techniques can be used to evaluate integrals involving radicals?
8. How do definite integrals relate to the area under a curve?

**Tip:** When dealing with integrals that involve square roots, it's crucial to handle the substitution step carefully to ensure the limits and the differential are correctly adjusted.

Learn how to solve the definite integral \( \int_{0}^{32} \left( \frac{1}{2} - 2\sqrt{2x} \right) dx \) with detailed steps and explanations. This integral involves the substitution method and integration of square roots. Suitable for advanced mathematics students and enthusiasts.

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