`int _(0) ^(a) 3x ^(2) dx = 8` then `a =

Options

A. 2
B. 0
C. `8/3`
D. a

Answer

To solve the integral equation \( \int_0^a 3x^2 \, dx = 8 \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_0^a 3x^2 \, dx \]
We know that this integral is equal to 8:
\[ I = 8 \]

### Step 2: Compute the integral
To compute the integral, we first find the antiderivative of \( 3x^2 \):
\[ \int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3 \]
Now we evaluate this from 0 to \( a \):
\[ I = \left[ x^3 \right]_0^a = a^3 - 0^3 = a^3 \]

### Step 3: Set the integral equal to 8
Now we set the result of the integral equal to 8:
\[ a^3 = 8 \]

### Step 4: Solve for \( a \)
To find \( a \), we take the cube root of both sides:
\[ a = \sqrt[3]{8} \]
Calculating the cube root gives us:
\[ a = 2 \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{2} \]