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

1. anonymous

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

2. anonymous

$$\dfrac{5}{n}$$ 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 $$k=3$$.