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(D) Case $I$: The area generated by the line segment $l$ is the curved surface area of a cylinder with radius $r$ and height $a$.$A_1 = 2 \pi r a$Case

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$A_1 = A_2 = A_3$

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(D) Case $I$: The area generated by the line segment $l$ is the curved surface area of a cylinder with radius $r$ and height $a$.$A_1 = 2 \pi r a$Case $II$: The area generated is the area of an annulus (ring) formed by rotating a line segment of length $a$ perpendicular to the axis $L$ at distance $r$ from its midpoint.The outer radius is $R = r + a/2$ and the inner radius is $r_{in} = r - a/2$.$A_2 = \pi R^2 - \pi r_{in}^2 = \pi (r + a/2)^2 - \pi (r - a/2)^2 = \pi (r^2 + ar + a^2/4 - (r^2 - ar + a^2/4)) = 2 \pi r a$Case $III$: The area generated is the lateral surface area of a frustum of a cone.The slant height is $l = a$. The radii of the two circular bases are $r_1 = r + (a/2) \sin 30^{\circ} = r + a/4$ and $r_2 = r - (a/2) \sin 30^{\circ} = r - a/4$.The lateral surface area of a frustum is $A_3 = \pi (r_1 + r_2) l = \pi (r + a/4 + r - a/4) a = \pi (2r) a = 2 \pi r a$.Thus,$A_1 = A_2 = A_3$.

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