Chapter 6, Problem 1RE

Describe the method of slicing for finding volumes, and use that method to derive an integral formula for finding volumes by the method of disks.

To determine

The method of slicing for finding volumes, and use that method to derive an integral formula for finding volumes by the method of disks.

Answer to Problem 1RE

The formula for volume by slicing method is $V={\displaystyle \underset{a}{\overset{b}{\int }}A\left(x\right)dx}$ .

The formula for volume by the method of disks is $V={\displaystyle \underset{a}{\overset{b}{\int }}\pi {\left[f\left(x\right)\right]}^2}dx$ .

Explanation of Solution

Slicing method:

In this method, the solid is divided into thin slabs. Then, the volume of each slab is approximated. The approximations are added to form a Riemann sum and the limit of the Riemann sum is taken to produce an integral for the volume.

Let $S$ be a solid that extends along the $x\text{-axis}$ and is bounded on the left and right sides by the planes that are perpendicular to the lines $x=a\text{ and }x=b$ .

Let the cross-section area $A\left(x\right)$ is defined for each $x$ in the interval $\left[a,b\right]$ .

Now, divide the solid into $n$ slabs.

Consider the width of $k\text{th}$ slab be $\Delta x_k$ and the volume of the $k\text{th}$ slab is approximated as $A\left(x_{{}_k}^{*}\right)\Delta x_k$ .

Where, $A\left(x_{{}_k}^{*}\right)$ is the cross-section area of the $k\text{th}$ slab with width or height $\Delta x_k$ .

The approximations are added to form a Riemann sum ${\displaystyle \sum _{k=1}^{n}A\left(x_{{}_k}^{*}\right)\Delta x_k}$ .

Take the limit of the Riemann sum to get the volume by slicing method as,

$\begin{array}{c}V=\underset{\max \Delta x_k\to 0}{\lim }{\displaystyle \sum _{k=1}^{n}A\left(x_{{}_k}^{*}\right)\Delta x_k}\\ ={\displaystyle \underset{a}{\overset{b}{\int }}A\left(x\right)dx}\end{array}$

Therefore, the formula for volume by slicing method is $V={\displaystyle \underset{a}{\overset{b}{\int }}A\left(x\right)dx}$ .

Finding volumes by the method of disks:

The method of finding the volumes of the solid with the disk-shaped cross sections is called the method of disks.

Let $f$ be a continuous and non-negative function on $\left[a,b\right]$ , and $R$ be the region bounded above by $y=f\left(x\right)$ , below by the $x\text{-axis}$ , and on the sides by the lines $x=a\text{ and }x=b$ .

Now, find the solid of revolution that is generated by revolving $R$ about the $x\text{-axis}$ .

The cross section of the solid taken perpendicular to the $x\text{-axis}$ at the point $x$ is a circular disk of radius $f\left(x\right)$ .

The area of the region is $A\left(x\right)=\pi {\left[f\left(x\right)\right]}^2$ .

Therefore, the formula for volume by the method of disks is $V={\displaystyle \underset{a}{\overset{b}{\int }}\pi {\left[f\left(x\right)\right]}^2}dx$ .