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Calculus/Volumes of Revolution

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Expert: - 5/5/2010


Question
As shown in the accompanying figure, a cocktail glass with a bowl shaped like a hemisphere of diameter 8cm contains a cherry with a diameter of 2cm. If the glass is filled to a depth of h cm, what is the volume of liquid it contains? (hint: first consider the case where the cherry is partially submerged, then the case where it is totally submerged)

additional hints:
-two integrals are needed
-hemisphere = half circle = disc method
-x-y grid
-Volume of a sphere: 4(pi)(r^3)

Unfortunately, I cannot upload the picture. It is a champagne glass that is an exact half-circle. the liquid is filled to slightly below the middle of the glass. a cherry is inside the glass. In one scenario, it is totally submerged in the liquid. In the other scenario, it is partially submerged.

Thank you so much for your help!

Answer
The cherry has diameter 2, so the radius is 1.  To talk about the cherry, for a sphere, the total volume is 4πr³/3.  Half the cherry volume is 2πr³/3.

In 3-dimensional spherical coordinates, that is r goes from 0 to √(16-z²),
z goes from -4 to -4+h, and Θ goes from 0 to 2π.

The integral is then ∫∫∫r dr dz dΘ, where the limits are given above.

If there's problem with that integral, feel free to write back.

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