If `A(x)=int_(0)^(x)t^(2)` dt, then : A (3) =

Options

A. 27
B. 3
C. 9
D. 81

Answer

To solve the problem, we need to evaluate the function \( A(x) \) defined as:

\[ A(x) = \int_{0}^{x} t^2 \, dt \]

We want to find \( A(3) \).

### Step 1: Evaluate the integral

To evaluate \( A(x) \), we first compute the integral:

\[ A(x) = \int_{0}^{x} t^2 \, dt \]

The integral of \( t^2 \) is:

\[ \int t^2 \, dt = \frac{t^3}{3} + C \]

### Step 2: Apply the limits of integration

Now we apply the limits from 0 to \( x \):

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

This simplifies to:

\[ A(x) = \frac{x^3}{3} \]

### Step 3: Substitute \( x = 3 \)

Now, we substitute \( x = 3 \) into the expression for \( A(x) \):

\[ A(3) = \frac{3^3}{3} \]

Calculating \( 3^3 \):

\[ 3^3 = 27 \]

Thus,

\[ A(3) = \frac{27}{3} = 9 \]

### Final Answer

Therefore, the value of \( A(3) \) is:

\[ \boxed{9} \]

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