`int_(0)^(a) y^(2) dy`

Answer

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

### Step 1: Identify the integral
We need to evaluate the definite integral of \( y^2 \) from 0 to \( a \).

### Step 2: Apply the power rule for integration
The power rule for integration states that:
\[ \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \]
In our case, \( n = 2 \). Therefore, we have:
\[ \int y^2 \, dy = \frac{y^{2+1}}{2+1} = \frac{y^{3}}{3} + C \]

### Step 3: Evaluate the definite integral
Now we will evaluate the definite integral from 0 to \( a \):
\[ \int_{0}^{a} y^{2} \, dy = \left[ \frac{y^{3}}{3} \right]_{0}^{a} \]
This means we will substitute \( a \) and \( 0 \) into \( \frac{y^{3}}{3} \).

### Step 4: Substitute the limits
Substituting the upper limit \( a \):
\[ \frac{a^{3}}{3} \]
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:
\[ \frac{a^{3}}{3} - 0 = \frac{a^{3}}{3} \]

Thus, the value of the definite integral \( \int_{0}^{a} y^{2} \, dy \) is:
\[ \frac{a^{3}}{3} \]

### Final Answer
\[ \int_{0}^{a} y^{2} \, dy = \frac{a^{3}}{3} \]

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