If `int_(0)^(a) 4x^(3) dx =16, and agt 0 , " then " a=`

Options

A. 0
B. -2
C. 2
D. `pm2`

Answer

To solve the problem, we need to evaluate the definite integral given and find the value of \( a \).

### Step-by-Step Solution:

1. **Set up the integral equation**:
We are given the equation:
\[ \int_{0}^{a} 4x^3 \, dx = 16 \]

2. **Factor out the constant**:
We can factor out the constant \( 4 \) from the integral:
\[ 4 \int_{0}^{a} x^3 \, dx = 16 \]

3. **Evaluate the integral**:
The integral of \( x^3 \) is:
\[ \int x^3 \, dx = \frac{x^4}{4} \]
Therefore, we can evaluate the definite integral:
\[ \int_{0}^{a} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{a} = \frac{a^4}{4} - \frac{0^4}{4} = \frac{a^4}{4} \]

4. **Substitute back into the equation**:
Now substituting this back into our equation:
\[ 4 \cdot \frac{a^4}{4} = 16 \]
This simplifies to:
\[ a^4 = 16 \]

5. **Solve for \( a \)**:
To find \( a \), we take the fourth root of both sides:
\[ a = \sqrt[4]{16} \]
Since \( 16 = 2^4 \), we have:
\[ a = 2 \]
Note that \( a \) can also be \( -2 \), but since we are given that \( a > 0 \), we take:
\[ a = 2 \]

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