1. The problem statement, all variables and given/known data You have a cylinder standing upright. Now, imagine a plane cutting the cylinder in half diagonaly from bottom left to top right. Develop a function for the volume of the half-cylinder (the top left piece) as a function of its height. Volume would be liquid rising from the bottom of the piece up to the top. The only variables I have are radius of the cylinder, and height of the cylinder. This isn't a homework problem... it's an oddly shaped tank I need to develop a curve for, but I figured this is the best place to post. 2. Relevant equations V = Pi*R^2*H -not very usefull A = 1/2 * R^2 * (Theta - Sin(theta)) -This is the area of the segment of a circle made by a chord, where theta is the angle formed by drawing two radii from the center of the circle to each end of the chord. 3. The attempt at a solution For a solution I'd really like a simple equation and/or something I can reference from a good source. The way I did it is the following (I'd like something more simple though if it's possible): Imagine the top view of the half-cylinder tank. As water rises from the bottom, you will see a cross sectional area fill that is equal to an area segment created by a chord. You can take that area and multiply it by a differential height then integrate over the height of the tank. So I take the equation for the area segment: A = (1/2) * R^2 * (Theta - Sin(Theta)) Theta would start at 0 and increase to 2Pi as the tank fills. However, I need to turn this equation into a function of height (y): Theta = 2*ArcCos[1-(2y/H)] where H = total height of the tank and y = height Plug in the new value for theta, and simplify: A = (1/2) * R^2 * ((2*ArcCos[1-(2y/H)]) - ((1-(2y/H))*(1-(1-(2y/H)^2)^.5)) To make this easier to read I'll consider Z = 1-(2y/H) and substitute it in (I can't use latex on this browser, sorry): A = (1/2) * R^2 * ((2*ArcCos[Z]) - (Z*(1-Z^2)^.5)) This equation can then be integrated from 0 up to y (where y would be the height of interest). Solving that integral analytically is very hard (impossible?). I used Excel to numerically integrate, but it is kind of sketchy. I'd like a more simple equation that gives an exact answer if it exists. Thanks for any help.