Mathematics

Problem

Consider the integral

\displaystyle \int_{0}^{1} 2x\,dx

. For each positive integer

n

, let the partition of

[0,1]

be defined by

x_i=\left(\dfrac{i}{n}\right)^2

for

i=0,1,\dots ,n

, and choose the right endpoint of each subinterval as the sample point. Which of the following limits equals the integral? a)

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\frac{2i(2i-1)}{n^{3}}

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\frac{2(i-1)^{2}(2i-1)}{n^{4}}

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\frac{2i^{2}(2i-1)}{n^{4}}

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\frac{2i^{2}}{n^{3}}