Chapter 5.5, Problem 6E

(a)

To determine

To find : The value of integral.

Answer to Problem 6E

The answer is: $T\approx 1.896$

Explanation of Solution

Given information:

The integration is ${{\displaystyle \int }}_0^{\pi }\sin xdx$

Calculation:

Let, $f(x)=\sin x$

Hence,

  $\begin{array}{c}n=4,\\ a=0,\\ b=\pi \end{array}$

Then,

  $\begin{array}{c}h=\frac{b-a}{n}\\ =\frac{\pi -0}{4}\\ =\frac{\pi }{4}\end{array}$

Fill out the table with $x$ and $f(x)$ :

$x$	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{3\pi }{4}$	$\pi$
$f(x)$	0	$\frac{\sqrt{2}}{2}$	1	$\frac{\sqrt{2}}{2}$	0

To calculate $T$ :

  $\begin{array}{c}\text{T}=\frac{\pi /4}{2}\left(0+2\times \frac{\sqrt{2}}{2}+2\times 1+2\times \frac{\sqrt{2}}{2}+0\right)\\ \approx 1.896\end{array}$

Therefore, the required value of integral is $T\approx 1.896$

(b)

To determine

To find: The function is an overestimate or underestimate.

Answer to Problem 6E

The answer is: Underestimate

Explanation of Solution

Given information:

The integration is ${{\displaystyle \int }}_0^{\pi }\sin xdx$

Calculation:

Hence, $f^{\prime }(x)=\cos x$

And

  $f^{″}(x)=-\sin x<0$

For

  $x\in [0,\pi ]$

Therefore, the required approximation is underestimate.

(c)

To determine

To find: The integral exact value.

Answer to Problem 6E

The answer is: $\text{2}$

Explanation of Solution

Given information:

The integration is ${{\displaystyle \int }}_0^{\pi }\sin xdx$

Calculation:

  $\begin{array}{c}{{\displaystyle \int }}_0^{\pi }\sin xdx=-{\left.\cos x\right|}_0^{\pi }\\ =-\cos \pi +\cos 0\\ =-(-1)+1\\ =2\end{array}$

Therefore, the required integral exact value is $\text{2}$ .