A = \int_{0}^{\fr{32}} \left( \frac{1}{2} - 2\sqrt{2x} \right) dx

To solve the integral \( A = \int_{0}^{\frac{3}{2}} \left( \frac{1}{2} - 2\sqrt{2x} \right) \, dx \), we will integrate each term separately.

### Step 1: Break the integral into parts
\[
A = \int_{0}^{\frac{3}{2}} \frac{1}{2} \, dx - \int_{0}^{\frac{3}{2}} 2\sqrt{2x} \, dx
\]

### Step 2: Integrate each term separately

#### First integral:
\[
\int_{0}^{\frac{3}{2}} \frac{1}{2} \, dx = \frac{1}{2} \int_{0}^{\frac{3}{2}} dx = \frac{1}{2} \left[ x \right]_{0}^{\frac{3}{2}} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}
\]

#### 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:
\[
\int \sqrt{u} \, du = \int u^{\frac{1}{2}} \, du = \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} u^{\frac{3}{2}}
\]
Substitute back \( u = 2x \):
\[
\frac{2}{3} (2x)^{\frac{3}{2}}
\]
Now, evaluate this from \( x = 0 \) to \( x = \frac{3}{2} \):
\[
\left[\frac{2}{3} (2x)^{\frac{3}{2}}\right]_{0}^{\frac{3}{2}} = \frac{2}{3} \left[(2 \cdot \frac{3}{2})^{\frac{3}{2}} - (2 \cdot 0)^{\frac{3}{2}}\right] = \frac{2}{3} \times (3)^{\frac{3}{2}} = \frac{2}{3} \times 3\sqrt{3} = 2\sqrt{3}
\]

### Step 3: Combine results
\[
A = \frac{3}{4} - 2\sqrt{3}
\]

Thus, the value of \( A \) is \( \boxed{\frac{3}{4} - 2\sqrt{3}} \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the geometric interpretation of this integral?
2. How do you calculate definite integrals with substitution?
3. Can you explain the power rule for integration?
4. What are common techniques for evaluating integrals involving square roots?
5. How does substitution simplify complex integrals?
6. What is the significance of the lower and upper limits in definite integrals?
7. How do you determine the area under a curve using integration?
8. What are other real-world applications of definite integrals?

**Tip:** When working with square roots in integrals, substitution often simplifies the expression, making the integration process more manageable.

Learn how to calculate the definite integral of (1/2 - 2sqrt(2x)) from 0 to 3/2 with step-by-step instructions. This problem involves integration techniques including substitution and the power rule. Suitable for college-level students studying integral calculus.

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