Mathematics

Problem

Consider the Riemann sum

T_{n}=\sum_{i=1}^{n} \sqrt{1+\dfrac{4i}{n}}\,\dfrac{4}{n}.

Assume

n

is a positive integer. (a) Compute the width

\Delta x

of each subinterval. (b) Determine the numerical values of the left endpoint

a

and the right endpoint

b

of the integral that this sum approximates. (c) Rewrite

T_{n}

as an expression involving a definite integral whose value it approximates as

n\to\infty