Chapter 5.1, Problem 23E

(a)

To determine

To find : The value of $S_8$ of the volume of the cylinder .

Answer to Problem 23E

The answer: $S_8\approx 120.951$ , underestimate

Explanation of Solution

Given information:

The graph is:

  

Calculation:

To use the equation $y=\pi \left(16-x^2\right)$ from the previous exercise

  $\begin{array}{c}a=0,\\ b=4\text{, }\\ n=8\end{array}$

calculate the right Riemann Sum using your calculator. This gives

  $S_8\approx 120.951$

It is an underestimation because the rectangles are outlined by the semicircle.

  

(b)

To determine

To find: $\left|V-S_8\right|$ as the percentage of $V$ .

Answer to Problem 23E

The answer: $10\text{\%}$

Explanation of Solution

Given information:

The graph is:

  

Calculation:

The exact volume is:

  $\begin{array}{c}V=\frac{1}{2}\frac{4}{3}\pi r^3\\ =\frac{1}{2}\frac{4}{3}\pi 4^3\\ =134.04128\end{array}$

So,

  $\begin{array}{c}\frac{\left|V-S_8\right|}{V}\approx =\frac{|134.04128-120.95132|}{134.04128}\\ =0.10\\ =10\text{\%}\end{array}$

Therefore the required $10\text{\%}$ of $\left|V-S_8\right|$ as the percentage of $V$ .