Chapter 6.1, Problem 20E

(a)

To determine

The sum S₈ of the volume of the cylinders.

Answer to Problem 20E

The value of S₈ is 146.08406

Explanation of Solution

Given information:

  

Formula used:

The volume of cylinder is given by

  $V=\pi r^{2}h$

Calculation:

Since hemisphere is approximated with cylinders.

So, volume of cylinder is given by

  $V=\pi r^{2}h$

Here,

  $\begin{array}{l}r=\sqrt{16-x^2}\\ h=\Delta x\end{array}$

So, use LRAM with

  $\begin{array}{l}f\left(x\right)=\pi {\left(\sqrt{16-x^2}\right)}^2\\ f\left(x\right)=\pi \left(6-x^2\right)\end{array}$

And interval [0, 4] which is partitioned into eight equal subintervals so, length of a subinterval is 0.5 so, substitutes the value of x with 0.5 differences in f(x) from 0, 0.5.... up to 3.5 and then add and multiply with 0.5 then it is equal to S₈

  $\begin{array}{l}{S}_8=0.5\pi \left[16+15.75+15+13.75+12+9.75+7+3.75\right]\\ S_8=146.08406\end{array}$

S₈ is an overestimate because each rectangle is above the curve.

Conclusion:

The value of S₈ is 146.08406

(b)

To determine

The percentage of V to the nearest percentage.

Answer to Problem 20E

The percentage error is 8.98%.

Explanation of Solution

Given information:

  

Formula used:

The Volume of hemisphere is $V=\frac{2}{3}\pi r^3$

Calculation:

Volume of hemisphere is calculated as

  $\begin{array}{l}V=\frac{2}{3}\pi r^3\\ =\frac{2}{3}\pi 4^3\\ V=134.0413\end{array}$

Now, for percentage error

  $\begin{array}{l}\frac{V-S_8}{V}\times 100=0.08986\times 100\\ =8.98\%\end{array}$

Conclusion:

The percentage error is 8.98%