### Home > CALC > Chapter 4 > Lesson 4.4.2 > Problem4-144

4-144.

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.
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, $f(x)$? 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
Substitute:
You probably want to factor out the $2$.

What represents the integrand, $f(x)$?
In order to identify the $f(x)$, 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 $n$ because we are evaluating values $x$ that are infinitely close together.
So $f(x) = (1 + 2x)^2$
Substitute:

What are the bounds?
The bounds are the smallest input and the largest input into the variable.
Since the integrand is $f(x) = (1 + 2x)^2$, where
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
So the bounds are form $0$ to $1$.
Substitute: