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4-154.

If $n$ is a positive integer, write a definite integral to represent $\lim\limits _ { n \rightarrow \infty } \frac { 2 } { n } [ ( 1 + \frac { 2 } { n } ) ^ { 2 } + ( 1 + \frac { 4 } { n } ) ^ { 2 } + \ldots + ( 1 + \frac { 2 n } { n } ) ^ { 2 } ]$.

You should recognize this Riemann sum as the Definition of an Integral.

$\lim \limits_{x\rightarrow \infty }(\text{Riemann Sum with }n\text{ rectangles})= \int_{a}^{b}f(x)dx.$

Thus, your job is to connect the given Riemann sum to the parts of an integral:

Start by answering these questions: What represents $dx$? What represents the integrand, $f\left(x\right)$? What are the bounds?

What represents $dx$?
$dx$ is the infinitely small widths of the infinitely many rectangles.
$dx$ can be represented by the expression.

$\lim \limits_{n\rightarrow \infty }\frac{1}{n}.$

Substitute:

$\lim \limits_{n\rightarrow \infty }2dx\left[\left(1+\frac{2}{n}\right)^{2}+\left(1+\frac{4}{n}\right)^{2}+...+\left ( 1+\frac{2n}{n} \right )^{2}\right]$

You probably want to factor out the 2

$2\lim \limits_{n\rightarrow \infty }dx\left[\left(1+\frac{2}{n}\right)^{2}+\left(1+\frac{4}{n}\right)^{2}+...+\left ( 1+\frac{2n}{n} \right )^{2}\right]$

What represents the integrand, $f\left(x\right)$?
In order to identify the $f\left(x\right)$, you need to find the variable, $x$.
Look for the part of the series that is changing: the numerators!
The first numerator is $2$, then $4$, then $6$, then ... $2n$
Now each variable is being divided by $n$

$\left (\text{or multiplied by } \frac{1}{n} \right )$

because we are evaluating values $x$ that are infinitely close together.
So $f\left(x\right) = \left(1 + 2x\right)^{2}$
Substitute:

$2\lim \limits_{n\rightarrow \infty }dxf(x)=2\lim \limits_{n\rightarrow \infty }(1+2x)^{2}dx$

What are the bounds?
The bounds are the smallest input and the largest input into the variable.
Since the integrand is $f\left(x\right) = \left(1 + 2x\right)^{2}$, where

$x=\frac{i}{n}$

Let's look at the smallest and largest value of $x$.
The smallest value of $x$ is

$\frac{1}{n}, \text{ and }\lim \limits_{n\rightarrow \infty }\frac{1}{n}=0$

The largest value of $x$ is

$\frac{n}{n}, \text{ and }\lim \limits_{n\rightarrow \infty }\frac{1}{n}=1$

So the bounds are form $0$ to $1$.
Substitute:

$\int_{0}^{1}(1+2x)^{2}dx$