Provide an appropriate response.
-What definite integral is represented by $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 4 i } { n } \right) ^ { 2 } \frac { 4 } { n }$ ? Use the Fundamental Theorem to evaluate the integral.
A) $\int _ { 0 } ^ { 1 } 4 x ^ { 2 } \mathrm { dx } = \frac { 4 } { 3 }$
B) $\int _ { 0 } ^ { 4 } x ^ { 2 } d x = \frac { 64 } { 3 }$
C) $\int _ { 1 } ^ { 4 } x ^ { 2 } d x = 21$
D) $\int _ { 0 } ^ { 4 } x ^ { 3 } d x = 64$

Question

Quizplus · Accepted Answer

Using the formula for the Riemann sum, we have:
\begin{align*}
\lim _ { n ightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 4 i } { n } ight) ^ { 2 } \frac { 4 } { n } &= \int _ { 0 } ^ { 4 } x ^ { 2 } d x \
&= \left[ \frac{x^3}{3} ight]_0^4 \
&= \frac{64}{3}.
\end{align*}
Thus, the correct choice is (B).

Provide an Appropriate Response lim⁡n→∞∑i=1n(4in)24n\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 4 i } { n } \right) ^ { 2 } \frac { 4 } { n }limn→∞​∑i=1n​(n4i​)2n4​

Provide an Appropriate Response $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( \frac { 4 i } { n } \right) ^ { 2 } \frac { 4 } { n }$