$$ \lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{2} \Delta x_{k} ;[4,13] $$ The limit, expressed as a definite integral, is $$ \int_{x^{2} d x} $$.

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To evaluate the limit

$$
\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{2} \Delta x_{k}
$$

as a definite integral, we need to recognize that this expression represents a Riemann sum. The Riemann sum approximates the area under the curve of the function $$ f(x) = x^2 $$ over the interval $$[4, 13]$$.

### Step 1: Identify the function and interval
The function we are dealing with is $$ f(x) = x^2 $$, and the interval is $$[4, 13]$$.

### Step 2: Set up the definite integral
As $$\Delta \rightarrow 0$$, the Riemann sum converges to the definite integral of the function over the specified interval:

$$
\int_{4}^{13} x^2 \, dx
$$

### Step 3: Calculate the definite integral
Now, we will compute the integral:

$$
\int_{4}^{13} x^2 \, dx
$$

To do this, we first find the antiderivative of $$ x^2 $$:

$$
\int x^2 \, dx = \frac{x^3}{3} + C
$$

Now we evaluate the definite integral from 4 to 13:

$$
\left[ \frac{x^3}{3} \right]_{4}^{13} = \frac{13^3}{3} - \frac{4^3}{3}
$$

Calculating $$ 13^3 $$ and $$ 4^3 $$:

$$
13^3 = 2197, \quad 4^3 = 64
$$

Now substituting these values into the integral:

$$
\frac{2197}{3} - \frac{64}{3} = \frac{2197 - 64}{3} = \frac{2133}{3}
$$

### Step 4: Final result
Now we simplify:

$$
\frac{2133}{3} = 711
$$

Thus, the limit expressed as a definite integral evaluates to:

$$
\int_{4}^{13} x^2 \, dx = 711
$$

Therefore, the final answer is:

$$
\boxed{711}
$$
The limit as a definite integral is $$ \int_{4}^{13} x^2 \, dx = 711 $$.

\[ \lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{2} \Delta x_{k} ;[4,13] \] The limit, expressed as a definite integral, is \( \int_{x^{2} d x} \).

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