How do you find the volume of a wine glass?

Question

the wine glass has a cross section of a parabola for example x=.5y^2

Answer 1

I would fill a measuring cup and see how much I need to pour in to fill the wine glass :P

But if you want to use math for some reason, you can take a triple integral in polar co-ordinates or do some tricks to simplify it down to a 2 dimensional problem.

At any rate, seems how you have no upper limit for y, the volume will be infinite as parabolas go on forever.

Answer 2

We will assume that the wine glass is round and that the height of the wine glass is h (the distance from the bottom of the parabola to the top of the glass).

Set up an element of volume and then integrate. The element will be cylindrical, perpendicular to the y-axis and have a thickness of dy. The radius of the element will be x, the distance from the y-axis to the side of the glass. It will then be integrated from y=0 to y=h.

dV = pi^x^2 dy = pi(0.25y^4)dy

Integrate to get: Volume = (1/20)pi*y^5 evaluated from y=0 to y=h
Volume = (pi/20)h^5

I forgot to mention, you need the height for a definite volume. Just plug it into the equation above and you willhave your volume.

Do a quick check (assuming a height).
Lets take a big wine glass and make h=4 so max radius is 8.
The volume of a cone that encloses the wine glass is pi*r^3*h/3. So volume = pi*8^2*4/3 = 268

So the volume of the wine glass must be less than this.

Volume of glass = (pi/20)4*5 = (pi/20)1024 = 160.85

Answer 3

Who in their right mind would take a cross section of a wine glass to find its volume? Isn't the glass useless now?

Fill it with water and pour it out into a graduated cylinder.