Final answer:
The question involves calculating the volume of solids formed by revolving an ellipse around the x-axis and y-axis. For the x-axis, we get a prolate spheroid and its volume is given by the formula V = 4/3πa³. For the y-axis, it is an oblate spheroid with volume V = 4/3πb³.
Step-by-step explanation:
The student's question involves calculating the volumes of solids of revolution, specifically revolving an ellipse around two separate axes to form different types of spheroids. The ellipse is described by the equation 36x² + 16y² = 576, which, when simplified, becomes x²/4 + y²/9 = 1. This represents an ellipse with semi-major axis a = 4 and semi-minor axis b = 3.
(a) If we revolve the upper half of the ellipse about the x-axis, we form a prolate spheroid. To calculate its volume, we use the formula for the volume of an ellipsoid V = 4/3πabc. Since we have a prolate spheroid (two equal semi-diameters), c = a, and the volume is V = 4/3πa³.
(b) For revolution around the y-axis, we form an oblate spheroid. Again using the formula for the volume of an ellipsoid, this time a = b, and the volume is V = 4/3πb³.