How do you do this question?

Step-by-step explanation:

The integral was obtained using cylindrical shells.  The volume of each shell is:

dV = 2π r h t

where r is the radius, h is the height, and t is the thickness.

The thickness is either dx or dy.  Looking at the integral, we can tell it is dy.  So the solid is revolved about the x-axis (or a line parallel to the x-axis).

The radius will be y.  So the height must be x = 9/(9+y²).

So the solid is obtained by rotating the region 0 ≤ x ≤ 9/(9+y²), 0 ≤ y ≤ 3 about the x-axis.

the image is given below

since the options are not given so we draw it from the equation

the given equation is

y = (x-1) (x+4)

y = y² + 3x - 4

the above equation is a quadratic equation which will have two roots. if we draw the graph for it, it will look like this