What’s the volume of the solid obtained by rotating the shaded region about the line y=3

Question

The area is the region bounded by x=8-2y² and x=1-(1/4)y²

Use whatever method is better. Disks/washers or shells.

Answer 1

The two graphs intersect at the points (0,2) and (0,-2). The easiest method to use here is the shell method with shells stemming from y=3. The integral expression for the volume is then,

2

2π ∫ (3-y)(8-2y² - 1 + (1/4)y²) dy

-2

Evaluating this integral yields, 112π

Answer 2

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