Volume with Cylindrical Shells Method, Integration, Formula, Examples, Problems - Calculus

The "volume with shells" method, also known as the shell method, calculates the volume of a solid of revolution by integrating the volume of infinitesimally thin cylindrical shells. The formula is \(V=\int _{a}^{b}2\pi rh\,dx\) (for vertical rotation) or \(V=\int _{a}^{b}2\pi rh\,dy\) (for horizontal rotation), where \(r\) is the radius, \(h\) is the height of the shell, and \(dx\) or \(dy\) is the thickness. This method is often easier to use than the disk/washer method when the axis of rotation is perpendicular to the axis of the representative rectangle.

💡How to use the shell method
• Visualize the solid: Imagine a region in the plane. When you revolve it around an axis, it forms a 3D solid. The shell method views this solid as being made of many thin cylindrical "shells" stacked together.
• Determine the axis of rotation:
⚬ If rotating around a vertical axis (like the y-axis), you will integrate with respect to \(x\). The radius and height will be functions of \(x\).
⚬ If rotating around a horizontal axis (like the x-axis), you will integrate with respect to \(y\). The radius and height will be functions of \(y\).
• Define the radius (\(r\)) and height (\(h\)):
⚬ Radius (\(r\)): This is the distance from the axis of rotation to the representative rectangle.
⚬ Height (\(h\)): This is the length of the representative rectangle, which is usually a function of the variable of integration. When revolving two functions, the height is the top function minus the bottom function (for vertical rotation) or the rightmost function minus the leftmost function (for horizontal rotation).
• Set up the integral: Use the formula, which is the integral of the volume of a single shell (\(2\pi rh\,dx\) or \(2\pi rh\,dy\)) from the lower limit (\(a\)) to the upper limit (\(b\)) of your region.
⚬ Vertical axis rotation: \(V=\int _{a}^{b}2\pi xf(x)\,dx\), where \(x\) is the radius and \(f(x)\) is the height.
⚬ Horizontal axis rotation: \(V=\int _{a}^{b}2\pi yf(y)\,dy\).
• Solve the integral: Calculate the definite integral to find the total volume.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link

💡Chapters:
00:00 Volume with the cylindrical shell method
01:26 Worked example

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