1. 3.9 Volumes about the y-axis
Volume = π x2.dy  a b a x
Find the volume of the solid when the curve y = x2 -1 is rotated about the y- axis from y = -1 to y = 3.
V = π x2.dy  a b -1 3 y x
y = x2 -1
 y + 1 = x2
V = π (y+ 1).dy  -1 3 y2 2
+ y -1 3
= π
= π[(9/2 +3) – (1/2 – 1)]
= 8π units3

2. 3.9 Volumes about the x-axisy x
Volume = π y2.dx  a b
Find the volume of the solid when the curve y = x2 is rotated about the x- axis from x = 0 to x = 2.
V = π y2.dx  a b
y = x2 2 y x
 y2 = x4
V = π (x4).dx  2 2 x5 5
= π
= π(32/5 – 0/2)
= 62/3π units3