Find the volume obtained when the shaded region is rotated
through 360' about the y-axis.​

Using the shell method, the volume is given by

That is, each cylindrical shell with radius x from the axis of revolution has height equal to the vertical distance between the upper curve (y = 2) and the lower curve (y = x ² + 1). Then the area of this shell is 2πrh = 2π x (2 - (x ² + 1)).

Computing the integral gives the volume,

this shows a reflection over the x-axis and is translated one unit down and 6 units to the right (x + 6, y - 1)

hope this !

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