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The solid is obtained by rotating the region 0 ≤ x ≤ 1/y², 1 ≤ y ≤ 4 about the line y = -3 using cylindrical shells.

Step-by-step explanation:

Cylindrical shells were used.  The volume of each shell is:

dV = 2π r h t

where r is the radius,

h is the height or width,

and t is the thickness.

The thickness of each shell is dy.  The axis of rotation is y = a, so the radius of each shell is r = y − a.  Since y + 3 is a factor of the function, a = -3.  That leaves 1/y² as the width of the shell.

From the limits of integration, we know the shells are between y=1 and y=4.  Therefore, the solid is obtained by rotating the region 0 ≤ x ≤ 1/y², 1 ≤ y ≤ 4 about the line y = -3.

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step-by-step explanation: