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

If

*n*is a positive integer, write a definite integral to represent. Homework Help ✎

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 ... 2*n*.

Now each variable is being divided by *n* .

because we are evaluating values *x* that are infinitely close together.

So *f*(*x*) = (1 + 2*x*)^{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 + 2*x*)^{2}, where

Let's look at the smallest and largest value of *x*.

The smallest value of *x* is

The largest value of *x* is

So the bounds are form 0 to 1.

Substitute: