How Do You Determine the Radius in Solids of Revolution Problems?

[GRB 080319B]

Is there a simple or generalized way (formula) to generate the radius of a solid of revolution? How does the orientation of the function relative to the axis of revolution affect the radius (radius= 4-f(x) or 4+f(x))? Why is the radius sometimes only x or y , and other times some other function? I consistently get the radius wrong in the solid of revolution problems with non-zero axis, and don't know if its a conceptual problem or that I just never learned the correct way to determine the radius. Thank you.

 

[HallsofIvy]

What axis is the curve being rotated about? The radius is the distance from that line, along a line perpendicular to it, to the curve. IF the curve is rotated around the x-axis, then that distance is the y-coordinate of a point on curve. If the curve is rotated around the y-axis then that distance is the x-coordinate.

If the curve is rotated around the line y= -4, the distance is, first 4 up to the x-axis, y=0, and than the y-axis of the point: y+ 4.

 

[GRB 080319B]

What axis is the curve being rotated about? The radius is the distance from that line, along a line perpendicular to it, to the curve. IF the curve is rotated around the x-axis, then that distance is the y-coordinate of a point on curve. If the curve is rotated around the y-axis then that distance is the x-coordinate.

If the curve is rotated around the line y= -4, the distance is, first 4 up to the x-axis, y=0, and than the y-axis of the point: y+ 4.

When you say the distance between the curve and axis of rotation is the y-coordinate, is the y in (y+4) equal to the function of the curve, or just the y variable, if it's on the curve? I seem to be having this problem understanding when to use just the variable or the function of the curve with work and hydrostatic force questions as well. Also, does it matter where the curve is relative to the axis of rotation and the x and y axes, or just the axis of rotation, when finding the radius? Is finding the radius just a matter of adding or subtracting the value of the axis of rotation depending on if the curve is "above" or "below" the axis of rotation wrt the x and y axes. I'm sorry if you have answered the question already in your reply, I'm just not sure if I'm understanding it completely. Thank you.

 

[HallsofIvy]

When you say the distance between the curve and axis of rotation is the y-coordinate, is the y in (y+4) equal to the function of the curve, or just the y variable, if it's on the curve? I seem to be having this problem understanding when to use just the variable or the function of the curve with work and hydrostatic force questions as well.

There is no difference. The variable "y" in a coordinate system, is, by definition, the distance from the x-axis to a given curve.

Also, does it matter where the curve is relative to the axis of rotation and the x and y axes, or just the axis of rotation, when finding the radius? Is finding the radius just a matter of adding or subtracting the value of the axis of rotation depending on if the curve is "above" or "below" the axis of rotation wrt the x and y axes.

It is a matter, as I said before, of distance from one point to another. How that is written in terms of x and y will, of course, depend on where the curve is relative to the x and y axes. There is no fixed rule, it depends on the geometry of the situation. When you are rotating around the x or y axes, or axes parallel to them, it is pretty simple and, just like in finding differences between points on an axis, a matter of adding or subtracting. If you were, for example, rotating a curve around the line y= x, that would be more difficult.

I'm sorry if you have answered the question already in your reply, I'm just not sure if I'm understanding it completely. Thank you.