Chapter 6, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. A region R is revolved about the y-axis to generate a solid S. To find the volume of S, you could either use the disk/washer method and integrate with respect to y or use the shell method and integrate with respect to x.
2. b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.
3. c. If water flows into a tank at a constant rate (for example 6 gal/min), the volume of water in the tank increases according to a linear function of time.

To determine

Whether the statement “A region R is revolved about the y axis to generate a solid S. To find the volume S, you could use either the disk/ washer method and integrate with respect to y or the shell method and integrate with respect to x.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The shell method is a method of finding volumes by decomposing a solid of revolution into cylindrical shells.

Suppose that a thin vertical strip is revolved about the y axis. An object of revolution (one that looks like a cylindrical shell or an empty tin can with the top and bottom removed) is obtained.

Then the resulting volume of the cylindrical shell is the surface area of the cylinder times the thickness of the cylindrical wall, which is shown below.

$V=2\pi {\displaystyle \underset{a}{\overset{b}{\int }}x\left[f\left(x\right)\right]dx}$, where x is the distance to the axis of revolution, $f\left(x\right)$ is the length and dx is the width.

Disk method models the resulting three dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness.

This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution.

$V={\displaystyle \underset{a}{\overset{b}{\int }}{\left[R\left(x\right)\right]}^2}dx$, where $R\left(x\right)$ is the length and dx is the width.

Thus, the given statement is true.

To determine

Whether the statement “Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

In order to find the displacement between two time periods, it is required to integrate the velocity between given time periods.

To evaluate the position at any time, that is either the initial position of the object at any given time or at least the position of the object at a particular time is required.

Thus, the given statement is true.

To determine

Whether the statement is “If water flows into a tank at a constant rate (for example 6gal/min.), the volume of water in the tank increases according to a linear function of time” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

If $\frac{dV}{dt}$ is constant, then V is linear function of time.

Here the rate of inflow is in units of volume per unit time.

Hence, the change in volume in the tank will also be linear.

Thus, the given statement is true.