`int _(0) ^(a) 3x ^(2) dx = 27` then `a = 2.5`. True or False.

Answer

To solve the integral \( \int_{0}^{a} 3x^2 \, dx = 27 \) and determine whether \( a = 2.5 \) is true or false, we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ \int_{0}^{a} 3x^2 \, dx \]

### Step 2: Integrate the function
The integral of \( 3x^2 \) can be computed using the power rule of integration:
\[ \int 3x^2 \, dx = 3 \cdot \frac{x^{3}}{3} = x^{3} \]
Thus, we have:
\[ \int_{0}^{a} 3x^2 \, dx = \left[ x^3 \right]_{0}^{a} = a^3 - 0^3 = a^3 \]

### Step 3: Set the integral equal to 27
Now, we set the result of the integral equal to 27:
\[ a^3 = 27 \]

### Step 4: Solve for \( a \)
To find \( a \), we take the cube root of both sides:
\[ a = \sqrt[3]{27} = 3 \]

### Step 5: Compare with the given value
The problem states that \( a = 2.5 \). Since we found \( a = 3 \), we can conclude that the statement is false.

### Final Conclusion
The statement that \( a = 2.5 \) is **False**.

---