The Dipstick Problem - area of a Cyclinder

[dipique]

There is a circle with a radius of 5 bisected by the y-axis. If we move it up and down the y-axis, different percentages of the circle will be above and below the x-axis. What I need to find out is how to relate the percentage below/above the x-axis (either would work) to the distance between the bottom of the circle and the x-axis.

So: we'll start with the circle sitting right on top of the x-axis. 100% of the circle area is above the x-axis. If we want to move it down so that 10% of the circle's area is below the x-axis, how many units would the circle have to be moved down?

Dan

 

[factor]

There are three special cases in this problem that I'll note first.

1) When a = 0, 50% of the area is above and 50% is below the x-axis
2) When a >= 5, 100% of the area is above the x-axis
3) When a <= -5, 100% of the area is below the x-axis

The next two cases are between these bounds of course.

When 0 < a < 5 the percentage below the x-axis is the integral from -sqrt(25 - a^2) to sqrt(25 - a^2) of -sqrt(25 - x^2)*dx divided by 25*Pi.

When -5 < a < 0 the percentage above the x-axis is the integral from -sqrt(25 - a^2) to sqrt(25 - a^2) of sqrt(25 - x^2)*dx divided by 25*Pi.