Calculus, volume of half sphere?

Obviously the volume of a hemisphere is half of the whole sphere. So I can show you how to find the volume of a sphere using one of my favorite techniques: Theorem of Pappus.

The theorem states roughly that if you rotate a non-overlapping area (with area A) around an axis that is distance d away from the center of the area, then the volume you create is A*d.

So for a sphere, imagine the top half of the circle x² + y² = r². We want to rotate this around the x-axis. This will create a sphere of radius r. We know the area of this half circle is A = ∫ dA = πr²/2. Now all we need to find is the center of this half circle. By symmetry, we know the center is at x=0. But we don't know where on the y-axis. If we knew, that would give us distance from center to x-axis. This value of y is called the y-centroid and is given by

y-centroid = (∫ y dA)/(∫ dA)

In our case,

d = y-centroid = (2/πr²) ∫_[-r, r] y dA

= (2/πr²) ∫_[-r, r] sqrt(r² - x²) dA

= (2/πr²) ∫_[-r, r] sqrt(r² - x²) (sqrt(r² - x²) dx)

= (2/πr²) ∫_[-r, r] r² - x² dx

= (2/πr²) ( r²x - x³/3 |_[-r, r]

= (2/πr²) ( 2r³/3 - (-2r³/3))

= (2/πr²) ( 4r³/3 )

= 8r/(3π)

So by Theorem of Pappus the volume of a sphere is πr²/2 * (8r/(3π)) = 4πr³/3. So the volume of a hemisphere is 2πr³/3.