Divanshu's question via email about cylindrical shells

In summary, the conversation discussed the process of finding the volume of a solid by rotating it around the y axis. This involves visualizing the solid as a collection of thin cylinders and using integration to find the exact volume. It was also mentioned that the method assumes the solid is continuous and has a smooth surface. Units of measurement should also be carefully considered in the calculation.
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Here is a sketch of the region to be rotated around the y axis.

View attachment 5639

You first need to visualise this entire region being rotated around the y axis, to get a mental picture of what the solid looks like. Then you need to imagine that the solid is made up of very thin vertically-oriented hollow cylinders. You can then approximate the volume of the solid by adding up the volumes of all the cylinders.

The curved surface of each cylinder is a rectangle. The width of each rectangle is the y value of the function. The length of the rectangle is the circumference of the cylinder, so $\displaystyle \begin{align*} 2\,\pi\,r \end{align*}$, and the radius of the cylinder is the x value of the function. So the area of each cylinder is $\displaystyle \begin{align*} 2\,\pi\,x\,y \end{align*}$, and thus the volume of each cylinder is $\displaystyle \begin{align*} 2\,\pi\,x\,y\,\Delta x \end{align*}$, where $\displaystyle \begin{align*} \Delta x \end{align*}$ is some small change in x.

So the total volume of the solid can be approximated by $\displaystyle \begin{align*} \sum{ 2\,\pi\,x\,y\,\Delta x } \end{align*}$.

If we increase the number of cylinders in the region and make each cylinder thinner, we get a better approximation of the total volume, so letting $\displaystyle \begin{align*} n \to \infty \end{align*}$ and $\displaystyle \begin{align*} \Delta x \to 0 \end{align*}$, the approximation becomes exact and the sum becomes an integral.

So the total volume of the solid is exactly

$\displaystyle \begin{align*} V &= \int_1^3{ 2\,\pi\,x\,y\,\mathrm{d}x } \\ &= 2\,\pi\int_1^3{ x\,\left( 2 + \frac{1}{5\,x} \right) \,\mathrm{d}x } \\ &= 2\,\pi \int_1^3{ \left( 2\,x + \frac{1}{5} \right) \,\mathrm{d}x } \\ &= 2\,\pi \,\left[ x^2 + \frac{x}{5} \right] _1^3 \\ &= 2\,\pi\,\left[ \left( 3^2 + \frac{3}{5} \right) - \left( 1^2 + \frac{1}{5} \right) \right] \\ &= 2\,\pi \,\left( 8 + \frac{2}{5} \right) \\ &= 2\,\pi\,\left( \frac{42}{5} \right) \\ &= \frac{84\,\pi}{5}\,\textrm{units}^3 \end{align*}$
 

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Thank you for sharing your calculation for the volume of the solid. Your approach of visualizing the solid as a collection of thin cylinders and then using integration to find the exact volume is a common and effective method in mathematics and physics. It is important to note that this method assumes the solid is continuous and has a smooth surface, which may not always be the case in real-world scenarios.

Additionally, as a scientist, it is important to consider the units of measurement in your calculation. In this case, the units of the volume should be in $\textrm{units}^3$, as you have correctly included in your final answer. It is also important to check the units of each term in your integral to ensure they are consistent.

Overall, your explanation and calculation are clear and well-reasoned. Keep up the good work!
 

FAQ: Divanshu's question via email about cylindrical shells

1. What are cylindrical shells and how are they used in science?

Cylindrical shells are three-dimensional structures that have a circular base and a curved surface that forms a cylinder. They are used in science for various purposes such as creating pressure vessels, storing liquids and gases, and constructing pipes and tubes.

2. How do cylindrical shells differ from other shapes?

Cylindrical shells have a curved surface and a circular base, whereas other shapes such as cubes and spheres have flat surfaces. This unique structure allows cylindrical shells to withstand external pressure and distribute weight evenly.

3. Can you explain the concept of stress and strain in relation to cylindrical shells?

Stress is the force per unit area that is applied to an object, while strain is the amount of deformation or change in shape that occurs in response to stress. In the case of cylindrical shells, the stress is distributed along the curved surface, resulting in minimal strain and high structural stability.

4. How are cylindrical shells used in engineering and construction?

Cylindrical shells are commonly used in engineering and construction for various applications such as building water tanks, silos, and pressure vessels. Their strong and stable structure makes them ideal for containing and transporting liquids and gases.

5. What factors can affect the strength of a cylindrical shell?

The strength of a cylindrical shell can be affected by several factors, including the material used, the thickness of the shell, the shape and size of the base, and the amount of stress and strain it is subjected to. It is important to carefully consider these factors when designing and constructing cylindrical shells for optimal strength and stability.

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