Chapter 5.5, Problem 20E

$\left(a\right)$

To determine

To find: The average value over the given interval using the trapezoidal sum.

Answer to Problem 20E

The approximate value of area under the curve is $50$ .

Explanation of Solution

Given information: Use trapezoidal sums to approximate the average value over the interval $\left[0,5\right]$ of a function with values given by the following table.

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$f(x)$	$4$	$7$	$8$	$6$	$4$	$2$

Using the trapezoidal rule for $n=5$ subintervals:

Here the subinterval width $\Delta x=1$ .

  $\begin{array}{l}{T}_5=\frac{△x}{2}\left[f\left(x_o\right)+2f\left(x_1\right)+2f\left(x_2\right)+2f\left(x_3\right)+2f\left(x_4\right)+f\left(x_5\right)\right]\\ =\frac{1}{2}\left[0+2\left(7\right)+2\left(8\right)+2\left(6\right)+2\left(4\right)+2\right]\\ =\frac{1}{2}\left[0+14+16+12+8+2\right]\\ =\frac{1}{2}\left[50\right]\\ =25\end{array}$

$\left(b\right)$

To determine

To find: The average of six values is less than the average value of the function over the interval $\left[0,5\right]$ .

Explanation of Solution

Given information: Consider the shape of a curve that would pass through these points. Explain why you would expect the average of these six values to be less than the average value of the function over the interval $\left[0,5\right]$ .

Since the trapezoidal rule evaluates the area under the curve by dividing the total area into small trapezoids. This rule takes the average of the left and the right sum.

Hence, the average of these six values is less than the average value of the function over the interval $\left[0,5\right]$ .