Why am I getting negative area?

[armolinasf]

Homework Statement

So, I have a cylinder with radius r and length L and it's filled with water that has a variable height H. I'm supposed to find a formula for the volume of the water contained in the cylinder in terms of H r and L.


3. Attempt at a solution
using a circle moved up on the y-axis by r units we get x^2+(y-5)^2=r^2

Then, integrating rectangles with width given by 2$$\sqrt{x^{2}-(y-r)^{2}}$$L$$\Delta$$y from 0 to H.

After evaluating it using trig substitution I get as my function :

V(H)=(L/2)((r^2-(H-r)^2)^(1/2)(H-r)+2r^2arcsin(H-r)/5)+pi*r^2)
This makes sense because when I let H=2r I get the volume of a cylinder and when H=0 I also get zero.

I derived the exact same formula using trigonometry and geometry as well.

My problem is that when I evaluate it for different height based on the diameters say H=(1/3)2r or H=(1/5)2r I get negative area which does not make sense.

I have a feeling that all of my (H-r) terms should be slightly altered, but I'm not sure, so if someone can point out what I'm missing it would be much appreciated

 

[Mark44]

Homework Statement

So, I have a cylinder with radius r and length L and it's filled with water that has a variable height H. I'm supposed to find a formula for the volume of the water contained in the cylinder in terms of H r and L.

How is the cylinder oriented? Is it lying on its side?

3. Attempt at a solution
using a circle moved up on the y-axis by r units we get x^2+(y-5)^2=r^2

Then, integrating rectangles with width given by 2$$\sqrt{x^{2}-(y-r)^{2}}$$L$$\Delta$$y from 0 to H.

Your formula for width should be in terms of one variable only. Since you are integrating with respect to y, the formula should only involve y.

To get y, solve the equation x² + (y - 5)² = r² for x.

After evaluating it using trig substitution I get as my function :

V(H)=(L/2)((r^2-(H-r)^2)^(1/2)(H-r)+2r^2arcsin(H-r)/5)+pi*r^2)
This makes sense because when I let H=2r I get the volume of a cylinder and when H=0 I also get zero.

I derived the exact same formula using trigonometry and geometry as well.

My problem is that when I evaluate it for different height based on the diameters say H=(1/3)2r or H=(1/5)2r I get negative area which does not make sense.

I have a feeling that all of my (H-r) terms should be slightly altered, but I'm not sure, so if someone can point out what I'm missing it would be much appreciated

Since you're getting a negative number, I suspect that a distance you are using turns out to be negative, possibly H - r.

 

[armolinasf]

The cylinder is lying on its side. My mistake with the variables, it should have been (r^-(h-r)^2)^(1/2). Would it matter if H is above or below the middle of the circle? I looked at my model and if H is below the middle of the circle I get the area that's not full.