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(A) We are given the definition of a definite integral as a limit of a sum: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n p} f\left(\frac{r}

Vedclass · Accepted Answer

$55$

Vedclass · Answer

(A) We are given the definition of a definite integral as a limit of a sum: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n p} f\left(\frac{r}{n}\right)=\int_0^p f(x) d x$. We need to evaluate $L = \lim _{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^{n} f\left(\frac{7k}{n}\right)$. Let $r = k$. The expression can be rewritten as $L = 3 \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(7 \cdot \frac{r}{n}\right)$. Let $g(x) = f(7x) = (7x)^2 + 2 = 49x^2 + 2$. Then the expression becomes $3 \int_0^1 g(x) dx = 3 \int_0^1 (49x^2 + 2) dx$. Evaluating the integral: $3 \left[ \frac{49x^3}{3} + 2x \right]_0^1 = 3 \left( \frac{49}{3} + 2 \right) = 49 + 6 = 55$. Thus, the correct option is $A$.

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