Calculate Volume of Cylindrical Shells

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In summary, the conversation discusses how to use cylindrical shells to calculate the volume of a solid formed by rotating the region bounded by y=x and y=x^2-2x about the line y=3. The speaker is unsure of how to write the equation in terms of x and asks for help setting up the problem. They also discuss the importance of making a sketch and choosing an appropriate coordinate system for integration.
  • #1
nobli1jl
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Use cylindrical shells to calculate the volume of the solid obtained by rotating the region bounded by y=x and y=x^2-2x about the line y=3
I know how do set the problem up and how to do it, I just don't know how to write "y=x^2-2x" in terms of "x". This is how I think the problem should be set up, with the changes made to the equation so everything in the equation will be "y's" or "dy's".

integrate from -1 to 0 : 2pi y (x^2-2x) dy + integrate from 0 to 3: 2pi y {y-(x^2-2x)} dy
 
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  • #2
Welcome to PF, nobli. We are happy to be of help, but you must show us your work to start things off. Have you made a sketch of the volume? What coordinate system would probably be the easiest to use for this integration? What is the form of the dV vector in that coordinate system?


EDIT -- fixed a typo
 

1. What is the formula for calculating the volume of cylindrical shells?

The formula for calculating the volume of cylindrical shells is V = πr²h, where V is the volume, r is the radius of the base, and h is the height of the shell.

2. How do you find the radius of a cylindrical shell?

To find the radius of a cylindrical shell, measure the distance from the center of the base to the edge of the base.

3. Can the volume of a cylindrical shell be negative?

No, the volume of a cylindrical shell can never be negative. It is a measure of the amount of space contained within the shell, and therefore must always be positive.

4. What units should be used for the measurements in the formula?

The units used for the measurements in the formula should be consistent. For example, if the radius is measured in meters, then the height should also be measured in meters.

5. How can the volume of cylindrical shells be applied in real life?

The concept of calculating the volume of cylindrical shells is often used in engineering and construction, for example in designing pipes or storage tanks. It can also be used in physics and chemistry to calculate the volume of test tubes or beakers.

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