Mathematics

Problem

For every positive integer

n

, let

R_n=\sum_{i=1}^{n} a\,x_i^{2}\,\Delta x,

where

x_i=1+\dfrac{4i}{n}

and

\Delta x=\dfrac{4}{n}

. Suppose

\displaystyle\lim_{n\to\infty} R_n=\int_{1}^{5} 2x^{2}\,dx

. (a) Express the limit of

R_n

as a definite integral in terms of

a

. Identify the integrand and limits of integration. (b) Calculate the value of

a

that guarantees

\displaystyle\lim_{n\to\infty} R_n=\int_{1}^{5} 2x^{2}\,dx

. Show all work leading to your result.