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

[JSM]

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?

 

[HallsofIvy]

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).

 

[JSM]

Thanks so much, sorry for the late reply. Wonderful explanation!