Key Concepts and Formulas

• Definite Integral as a Limit of a Sum (Riemann Sum): A definite integral can be represented as the limit of a sum. The fundamental formula is: $\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=a}^{b} f\left(\frac{r}{n}\right) = \int_0^1 f(x) \, dx$ More generally, for a sum of the form $\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=m}^{k} f\left(c + \frac{r}{n}\right)$, it can be converted to $\int_{c + m/n}^{c + k/n} f(x) \, dx$ as $n \to \infty$.
• Integral Evaluation: Using the power rule for integration: $\int u^k \, du = \frac{u^{k+1}}{k+1} + C$.

Step-by-Step Solution

Step 1: Analyze the Given Expression and Convert to Standard Summation Form

The given limit is: $L = \lim_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ Our goal is to express the sum inside the curly braces in sigma notation. Let's examine the terms: The first term is $4$. We can write $4 = 2^2 = \left(2+\frac{0}{n}\right)^2$. This corresponds to the case when $r=0$. The subsequent terms are of the form $\left(2+\frac{r}{n}\right)^2$ for $r=1, 2, \ldots$. The last term is $\left(3-\frac{1}{n}\right)^2$. We need to express this in the form $\left(2+\frac{r}{n}\right)^2$. $3 - \frac{1}{n} = 2 + 1 - \frac{1}{n} = 2 + \frac{n-1}{n}$ So, the last term is $\left(2+\frac{n-1}{n}\right)^2$. This corresponds to the case when $r=n-1$.

Thus, the sum inside the curly braces can be written as: $\sum_{r=0}^{n-1} \left(2+\frac{r}{n}\right)^2$ Now, substitute this back into the limit expression: $L = \lim_{n \rightarrow \infty} \frac{3}{n} \sum_{r=0}^{n-1} \left(2+\frac{r}{n}\right)^2$ We can factor out the constant $3$: $L = 3 \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{n-1} \left(2+\frac{r}{n}\right)^2$ This expression is now in the standard form for conversion to a definite integral.

Step 2: Convert the Limit of Sum to a Definite Integral

We use the definition of the definite integral as the limit of a Riemann sum. We identify the following:

• The term $\frac{r}{n}$ is replaced by $x$.
• The term $\frac{1}{n}$ is replaced by $dx$.
• The function $f(x)$ is determined from the general term $\left(2+\frac{r}{n}\right)^2$. Thus, $f(x) = (2+x)^2$.

Now, we determine the limits of integration:

• The sum starts with $r=0$. The lower limit of integration is $\lim_{n \rightarrow \infty} \frac{0}{n} = 0$.
• The sum ends with $r=n-1$. The upper limit of integration is $\lim_{n \rightarrow \infty} \frac{n-1}{n} = \lim_{n \rightarrow \infty} \left(1 - \frac{1}{n}\right) = 1$.

Therefore, the limit of the sum can be converted to the definite integral: $L = 3 \int_0^1 (2+x)^2 \, dx$

Step 3: Evaluate the Definite Integral

We now evaluate the definite integral: $L = 3 \int_0^1 (2+x)^2 \, dx$ Let $u = 2+x$. Then $du = dx$. When $x=0$, $u=2$. When $x=1$, $u=3$. $L = 3 \int_2^3 u^2 \, du$ Using the power rule for integration: $L = 3 \left[ \frac{u^{2+1}}{2+1} \right]_2^3$ $L = 3 \left[ \frac{u^3}{3} \right]_2^3$ The constant $3$ cancels out: $L = \left[ u^3 \right]_2^3$ Now, apply the limits of integration: $L = (3)^3 - (2)^3$ $L = 27 - 8$ $L = 19$

Common Mistakes & Tips

• Incorrectly identifying the summand: Ensure the entire expression inside the summation is correctly represented as $f(\frac{r}{n})$ or $f(c + \frac{r}{n})$. Pay close attention to algebraic manipulations, especially for the first and last terms.
• Errors in determining integration limits: The limits of integration are derived from the minimum and maximum values of $r$ divided by $n$ as $n \to \infty$.
• Forgetting constant factors: Any constant multiplier outside the summation and limit, such as the $\frac{3}{n}$ in this problem, must be carried through the conversion to the definite integral.

Summary

The problem involves evaluating a limit of a sum, which can be converted into a definite integral using the definition of a Riemann sum. By carefully rewriting the given sum in the standard sigma notation and identifying the function and integration limits, we transformed the limit into a definite integral. Evaluating this integral yielded the final result.

The final answer is $\boxed{19}$, which corresponds to option (C).