### Home > CALC > Chapter 2 > Lesson 2.1.3 > Problem2-40

2-40.

For the given function, write an expression in summation notation that will approximate $A(f, 3 ≤ x ≤ 7)$ using eight rectangles. Specify if you use left, right, or midpoint rectangles. Then enter this summation expression into your graphing calculator and evaluate the approximate area.

$f ( x ) = \frac { 2 ( x + 4 ) } { x + 6 }$

Recall problem 2-17:

i. For left-endpoint and midpoint rectangles, let the index start at $0$ and end at $n-1$.
For right-endpoint rectangles, let the index start at $1$ and end at $n$.
$n =$ number of rectangles.

ii. The $\frac{1}{2}$ represents the WIDTH/BASE of all rectangles.
To determine the width of each rectangle, divide the domain by the number of rectangles:
$\frac{7-3}{1}=\frac{1}{2}.$ Each rectangle has a width, $\Delta x \text{ is } \frac{1}{2}.$

iii. The transformed function represents the HEIGHT of each rectangle.
For left- and right-endpoint rectangles, the heights are represented by $f(a + Δxi)$.
For midpoint rectangles, the heights are represented by $f (a+\frac{\Delta x}{2}i).$
$a =$ starting value
$Δx =$ width of each rectangle.