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

3.9 Volumes about the x-axis y 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