How to set up a volume of solid of revolution about a line other than the x axis

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SUMMARY

This discussion focuses on calculating the volume of a solid of revolution around a line, specifically the line defined by the equation y = x/4. The correct method involves determining the radius of rotation, which is the distance from the function f(x) to the line y = x/4. The formula for volume is derived from the integral of the square of this radius, expressed as v = π * integral((f(x) - (x/4))^2)dx. Additionally, for slant lines, one must calculate the perpendicular distance from the line to the function to find the correct volume.

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  • Understanding of calculus, specifically integration techniques.
  • Familiarity with the concept of solids of revolution.
  • Knowledge of line equations and their geometric interpretations.
  • Ability to derive and manipulate functions to find intersections.
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  • Study the method of calculating volumes of solids of revolution using the washer method.
  • Learn about the shell method for finding volumes around vertical and slant lines.
  • Explore the concept of parametric equations in relation to solids of revolution.
  • Investigate the application of integration in real-world volume problems.
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JSM
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Hello folks, I was wondering how to set up a volume of the solid of revolution about a line in the form of a line equation. if i wanted to find the volume about a line of x/4 would I simply find it as v=pi*integral (f(x/4)^2)dx or is there a method I'm missing all togeather?
 
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No, that won't do it. In particular, you should be able to see that you do not replace "x" with "x/4" which is a y value. You need to recalculate "r". If, for example, you were to rotate y= f(x) around the x axis, you would analyse that by noting that each point on y= f(x) rotates around the x-axis in a circle with radius f(x). If you were rotating around some horizontal line, say y= a, with a< 0, then the distance from y= a to y= f(x) would be f(x)- a and you would use (f(x)-a)2.

If you have a slant line, like y= x/4, you need to calculate the distance from that line to a point on the graph along a line perpendicular[/itex] to that. In particular, for any given x0, the line through y= x/4 at (x0,x0/4) perpendicular to it is y= -(4/x0(x- x0= -4x/x0+ 4. You would need to determine where that intersect the graph y= f(x).
 
Thanks so much, sorry for the late reply. Wonderful explanation!
 

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