`int_(0)^(15) [x^(2)]dx` is equal to

Options

A. 2
B. `2-sqrt(2)`
C. `2+sqrt(2)`
D. `sqrt(2)`

Explanation

To solve the integral \( \int_{0}^{1.5} x^2 \, dx \), we can follow these steps:

### Step 1: Set up the integral
We need to evaluate the definite integral from 0 to 1.5 of the function \( x^2 \).

\[ \int_{0}^{1.5} x^2 \, dx \]

### Step 2: Find the antiderivative
The antiderivative of \( x^2 \) is given by:

\[ \frac{x^3}{3} \]

### Step 3: Evaluate the definite integral
Now we will evaluate the antiderivative at the upper limit (1.5) and the lower limit (0):

\[ \left[ \frac{x^3}{3} \right]_{0}^{1.5} = \frac{(1.5)^3}{3} - \frac{(0)^3}{3} \]

Calculating \( (1.5)^3 \):

\[ (1.5)^3 = 1.5 \times 1.5 \times 1.5 = 3.375 \]

Thus, we have:

\[ \frac{(1.5)^3}{3} = \frac{3.375}{3} = 1.125 \]

### Step 4: Final result
Since the lower limit contributes 0, the value of the definite integral is:

\[ \int_{0}^{1.5} x^2 \, dx = 1.125 \]

### Summary of the solution:
The value of \( \int_{0}^{1.5} x^2 \, dx \) is \( 1.125 \).

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