`int_(0)^(2) [x^(2)]dx` is equal to

Options

A. `5-sqrt(2)`
B. `5-sqrt(3)-sqrt(2)`
C. `5-sqrt(3)`
D. none of these

Explanation

To solve the integral \( \int_{0}^{2} x^2 \, dx \), we will follow these steps:

### Step 1: Set up the integral
We need to evaluate the definite integral:
\[ \int_{0}^{2} x^2 \, dx \]

### Step 2: Find the antiderivative
The antiderivative of \( x^2 \) is:
\[ \frac{x^3}{3} \]

### Step 3: Evaluate the definite integral
Now we will evaluate the antiderivative from 0 to 2:
\[ \left[ \frac{x^3}{3} \right]_{0}^{2} \]

### Step 4: Substitute the limits
Substituting the upper limit (2):
\[ \frac{2^3}{3} = \frac{8}{3} \]
Now substituting the lower limit (0):
\[ \frac{0^3}{3} = 0 \]

### Step 5: Calculate the result
Now we subtract the lower limit from the upper limit:
\[ \frac{8}{3} - 0 = \frac{8}{3} \]

Thus, the value of the integral \( \int_{0}^{2} x^2 \, dx \) is:
\[ \frac{8}{3} \]

### Final Answer
\[ \int_{0}^{2} x^2 \, dx = \frac{8}{3} \]

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