Question

How do you do this question?

Answer 1

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

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.

Answer 2

Your solution is the third option. Check the explanation in the attachment.