Q. 1 - Calculus

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

1. True or False: The thickness of a shell with inner radius $r$ and outer radius $R$ is $R-r$.
2. True or False: The average radius of a shell with inner radius $r$ and outer radius $R$ is $\frac{r+R}{2}$.
3. True or False: A shell with height $h$ and average radius $r$ has volume $\pi r^{2}h\Delta x$.
4. True or False: A shell with height $h$, average radius $r$, and thickness $\Delta x$ has volume $\pi r^{2}h\Delta x$.
5. True or False: A shell with height $h$, outer radius $r$, and thickness $\Delta x$ has volume $2\pi rh\Delta x$.
6. True or False: Rotating a vertical rectangle around the $y$-axis results in a shell.
7. True or False: Rotating a horizontal rectangle around the $y$-axis results in a shell.
8. True or False: Rotating a horizontal rectangle around the $x$-axis results in a shell.

Step-by-Step Solution

Answer

1. True
2. True
3. False
4. False
5. False
6. True
7. False
8. True

Step 1: Given information

1. The thickness of a shell with inner radius $r$ and outer radius $R$ is $R-r$.
2. The average radius of a shell with inner radius $r$ and outer radius $R$ is $\frac{r+R}{2}$.
3. A shell with height $h$ and average radius $r$ has volume $\pi r^{2}h\Delta x$.
4. A shell with height $h$, average radius $r$, and thickness $\Delta x$ has volume $\pi r^{2}h\Delta x$.
5. A shell with height $h$, outer radius $r$, and thickness $\Delta x$ has volume $2\pi rh\Delta x$.
6. Rotating a vertical rectangle around the $y$-axis results in a shell.
7. Rotating a horizontal rectangle around the $y$-axis results in a shell.
8. Rotating a horizontal rectangle around the $x$-axis results in a shell.

Step 2: Determine whether the statement is True or False.

1. A shell with inner radius $r$, outer radius $R$, and height $h$ is said to have thickness $∆x=R-r$.
2. A shell with inner radius $r$, outer radius $R$,, and height $h$ is said to have average radius $r_{av}=\frac{r+R}{2}$.
3. A shell with inner radius $r$, outer radius $R$, and height $h$ is said to have average radius $r_{av}=\frac{r+R}{2}$ and thickness $\Delta x=R-r$. Such a shell has volume $V=2\pi rh\Delta x$.
4. A shell with inner radius $r$, outer radius  $R$, and height $h$ is said to have average radius $r_{av}=\frac{r+R}{2}$ and thickness $\Delta x=R-r$. Such a shell has volume $V=2\pi rh\Delta x$.
5. A shell with inner radius $r$, outer radius $R$, and height $h$ is said to have average radius $r_{av}=\frac{r+R}{2}$ and thickness $∆x=R-r$. Such a shell has volume $V=2\pi rh\Delta x$.
6. When a vertical rectangle is rotated around the $y$-axis a cylinder is formed with height equal to the length of the vertical rectangle and radius half of the width of the vertical rectangle.
7. When a horizontal rectangle is rotated around the $y$-axis a disc is formed with radius equal to the width of the horizontal rectangle, and height equal to the length of the horizontal rectangle.
8. When a horizontal rectangle is rotated around the $x$-axis a cylinder is formed with height equal to the length of the vertical rectangle and radius half of the width of the horizontal rectangle.

Step 3: Write the conclusion

1. The given statement is True.
2. The given statement is True.
3. The given statement is False.
4. The given statement is False.
5. The given statement is False.
6. The given statement is True.
7. The given statement is false.
8. The given statement is True.