Question
The base of solid is right triangle whose base side has length a and whose perpendicular side has length (1/2)a. Find the volume of the solid if cross sections perpendicular to the base the triangle are semicircles.
Answers
oobleck
The hypotenuse of the triangle is the line y = (a-x)/2
That means the diameter of the semicircle at (x,y) is y, so the area is
π/4 y^2 = π/4 (a-x)^2/4
The volume is thus
v = ∫[0,a] π/16 (a-x)^2 dx
That means the diameter of the semicircle at (x,y) is y, so the area is
π/4 y^2 = π/4 (a-x)^2/4
The volume is thus
v = ∫[0,a] π/16 (a-x)^2 dx