The portion of the ellipse x^2/9+y^2/4=1 with x greater than or equals to 0 is rotated about the y-axis to form a solid S. A hole of radius 1 is drilled through the center of S, along the y-axis. Find the exact volume of the part of S that remains. Show all steps. (Hint: use shells)

Answer 1

Think of the shells as nested cylinders, starting 1 unit away from the y-axis, and extending to the end of the ellipse:

V = ∫[1,3] 2πrh dx
where r=x and h=y
V = 2π∫[1,3] x(2√(1-x^2/9)) dx
= 2π/3 ∫[1,3] 2x√(9-x^2) dx
= 2π/3 (32/3 √2)
= 64/9 √2 π

Answer 2

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