# Radius of a Solid of Revolution

## Main Question or Discussion Point

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
Homework Helper
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.

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