Chapter 5.5, Problem 21E

(a)

To determine

To explain: Double the value of endpoints two and get the average value of function as $6.2$ .

Answer to Problem 21E

The approximate value of area under the curve is $6.2$ .

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.

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

Consider the given information,

Now, add double two end points and find the average value of the function.

  $\begin{array}{c}\text{avg}f\left(x\right)=\frac{4+7+8+6+2\left(4\right)+2\left(2\right)}{6}\\ =\frac{4+7+8+6+8+4}{6}\\ =\frac{37}{6}\\ =6.1666\\ \approx 6.2\end{array}$

Therefore, the obtained value is $6.2$ .

(b)

To determine

The explanation about to makes more sense not to double the endpoint values.

Answer to Problem 21E

Due to increasing error, the doubling of the values does not make sense.

Explanation of Solution

Given information:

Refer the table form the previous question.

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

Consider the given information,

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.

While doubling the values of the table only endpoints, the error of first and last trapezoid will increase in the given interval $\left[0,5\right]$ .

Therefore, it is not make sense to double the endpoints values.