Remaining Volume?

Question

The question is as follows:
A cylindrical glass of radius r and heigh h is filled with water then titled until the water remaining in the glass exactly covers the bottom (circular base) of the glass. Determine the volume of the remaining water by slicing.

Work accompanying the answer would greatly be appreciated.

The answer is listed as (1/2)πr^2...

The work must be done using integrals....

Answer 1

Let r be the radius and h be the height of the cylinder.

Let A = f(x) be the cross sectional area at x, x from 0.5 h to h.

By symmetry, we have

volume
= ∫pi r^2 - f(h-x) dx, x from 0 to 0.5h
+∫f(x)dx, x from 0.5h to h
= (1/2)pi r^2 h
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Reason:
∫f(h-x) dx, x from 0 to 0.5h
= -∫f(u) du, u from h to 0.5h, where u = h-x
= ∫f(u) du, u from 0.5h to h
= ∫f(x) dx, x from 0.5h to h

Answer 2

No work needed:

Since the water slices the cylinder at 2 points which are diagonally opposite AND the object in question is symmetrical, it must be 1/2 original volume