Posted by Johnathon on Thursday, November 10, 2011 at 12:09pm.
I may have this wrong, but here's how I see it:
Draw a circle of radius 2r. That's the base. Draw a chord perpendicular to the x-axis at distance x from the center. At this point,
x^2 + y^2 = 4r^2
Now, erect a square of height 2y on the base. That is a cross-section of the solid. It has area 4y^2
The volume is thus Int(4y^2 dx)[0,2r]
But, y^2 = 4r^2 - x^2
v = 4*Int(4r^2 - x^2) dx)[0,2r]
= 4*(4r^2 x - 1/3 x^3)[0,2r]
= 4(8r^3 - 8r^3/3)
= 32(2/3 r^3)
= 64/3 r^3
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