Write the integral that produces the same value as\lim_{n \rightarrow \infty } \sum_{i=1}^{n}(3+i(5/n))^2)(5/n)

Question

anonymous · Answer

$$\lim_{n \rightarrow \infty}\sum_{i=1}^{n}(3+i(5/n))^2(5/n)$$

anonymous · Answer

$\dfrac{5}{n}$ suggests we're approximating an integral with $n$ rectangles over the interval $[k,k+5]$, since $\dfrac{5}{n}=\dfrac{k+5-k}{n}$.

Next, the fact that we're evaluating the square of some expression suggests, we're approximating the definite integral of $f(x)=x^2$ over the interval above.

When $i=1$, you have the value of $x$ is $3+\dfrac{5}{n}$, which approaches $3$ as $n\to\infty$. This suggests the start of the interval is $k=3$.