`int_(0)^(2) sqrt(2-x)dx`.

Answer

To solve the integral \( I = \int_{0}^{2} \sqrt{2 - x} \, dx \), we will use the substitution method. Here’s a step-by-step solution:

### Step 1: Make a substitution
Let \( t = 2 - x \). Then, we differentiate to find \( dx \):
\[ dx = -dt \]

### Step 2: Change the limits of integration
When \( x = 0 \):
\[ t = 2 - 0 = 2 \]
When \( x = 2 \):
\[ t = 2 - 2 = 0 \]
Thus, the limits change from \( x: 0 \to 2 \) to \( t: 2 \to 0 \).

### Step 3: Substitute into the integral
Now, substituting \( t \) and \( dx \) into the integral, we get:
\[ I = \int_{2}^{0} \sqrt{t} \cdot (-dt) = \int_{0}^{2} \sqrt{t} \, dt \]

### Step 4: Evaluate the integral
The integral \( \int \sqrt{t} \, dt \) can be computed as follows:
\[ \int \sqrt{t} \, dt = \frac{t^{3/2}}{3/2} = \frac{2}{3} t^{3/2} \]
Now, we evaluate this from \( 0 \) to \( 2 \):
\[ I = \left[ \frac{2}{3} t^{3/2} \right]_{0}^{2} = \frac{2}{3} \left( 2^{3/2} - 0^{3/2} \right) \]

### Step 5: Calculate \( 2^{3/2} \)
Calculating \( 2^{3/2} \):
\[ 2^{3/2} = (2^3)^{1/2} = \sqrt{8} = 2\sqrt{2} \]
Thus, we have:
\[ I = \frac{2}{3} (2\sqrt{2}) = \frac{4\sqrt{2}}{3} \]

### Final Answer
The value of the integral \( I \) is:
\[ \boxed{\frac{4\sqrt{2}}{3}} \]