### Home > APCALC > Chapter 5 > Lesson 5.4.1 > Problem5-147

5-147.

A function $f$ is continuous over the interval $[2, 8]$ and has values that are given in the table at right.

1. Using the sub-intervals $[2, 5], [5, 7]$, and $[7, 8]$, what is the trapezoidal approximation of $\int _ { 2 } ^ { 8 } f ( x ) d x$?

The Trapezoid Rule

The Trapezoid Rule can be used to approximate the area under a curve if the widths, $\Delta x$, of all trapezoids are of equal size:

$\left. \begin{array} { r } { A = \frac { \Delta x } { 2 } ( f ( a ) + 2 f ( a + \Delta x ) + 2 f ( a + 2 \Delta x ) + \ldots } \\ { + 2 f ( b - \Delta x ) + f ( b ) ) } \end{array} \right.$

2. Over which sub-interval is the average slope a minimum?

1. $[2, 5]$

1. $[5, 7]$

1. $[7, 8]$

Test all answer choices and choose the lowest value. Recall that average slope is akin to saying average rate of change (AROC).

$x$

$f \left(x\right)$

$2$

$10$

$5$

$30$

$7$

$40$

$8$

$20$