The base of a solid is the region in the first quadrant bounded by the line x+2y=4 and the coordinate axes, what is the volume of the solid if every cross section perpendicular to the x-axis is a semicircle?

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Answer 1 · 2016-07-10T19:58:18

#= 2/3 pi#

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drawing refers

the cross section area of the solid at point x, given it is a semi circle, on the x axis is

#A(x) = (pi y^2)/8#, see drawing

So the volume of the solid is:

#V = int_0^4 \ A(x) \ dx#

#= int_0^4 \ (pi y^2)/8\ dx#

#= pi/8 int_0^4 \ ((4-x)/2)^2\ dx#

#= pi/32 int_0^4 \ (4-x)^2\ dx#

#= pi/32 * 64/3#

#= 2/3 pi#