How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations for

$21$

$$

$1$x

$$dx

are accurate to within $0\mathrm{.}00002?$

Solution

If

f$($x$)=$

$1$x

$,$

then

f$($x$)=$

$,$

and

f$($x$)=$

$.$

Since

$1$ x $2,$

we have

$1$x

$1,$

so

$|$f$($x$)|=$

$$

$213$

$=.$

We take K $=2,$ a $=1,$ and b $=2.$ Accuracy to within $0\mathrm{.}00002$ means that the magnitude of the error should be less than $0\mathrm{.}00002.$ Therefore, we chose n so that

$2$

$3$

$12$n$2$

$<0\mathrm{.}00002.$

Solving the inequality for n$,$ we get

n$2>$

$2$

$3$

$12(0\mathrm{.}00002)$

or

n $>$

$1$

$0\mathrm{.}00012$

$91\mathrm{.}29.$

Thus, n $($rounded up to the nearest integer$)$ is and will ensure the desired accuracy.

o achieve the same accuracy using the Midpoint Rule we choose n so that

$2$

$3$

$24$n$2$

$<0\mathrm{.}00002,$

which gives the following. $($Round your answer up to the nearest integer.$)$

n $>$

$1$

$0\mathrm{.}00024$

$$