Find the volume of the solid formed by revolving the given regions about the y-axis.?

Question

a) the region bounded by the curve y = (x square) , and the lines x = 0 and y = 3

b) the region bounded by the parabola y = 9 - x^2 , and y = 0 (first quadrant only)

Answer 1

a/ y = x²

Volume and the y-axis
V = pi ⌡ x² dy where pi = 3.14 (2dp.)

from y = 0 to y = 3 make x the subject : x = √y

V = pi⌡ y dy 0 to 3
= pi (1/2) [y² ] 0 to 3
= pi (1/2) (9)
= pi x 9/2
= 14.13 units cubed (2.dp)

b/ y=0 y=8 x = (9-y)^1/2

V = pi ⌡ 9 - y dy 0 to 3
= pi [ 9x - (1/2) x^2 ] 0 to 3
= pi ( 27 - (1/2) 3 - 9 + 0 )
= pi 16.5
= 51.81 units cubed (2dp)

Answer 2

using cylindrical coordinates about the y-axis,
dV = rdrdθdy
a)
V = ∫∫∫rdrdθdy
V = 2π∫∫rdrdy
V = π∫x²dy
V = π∫ydy
V = 9π/2

b)
x² = 9 - y
V = π∫(9 - y)dy
V = π(9y - y²/2)
V = π(81 - 81/2)
V = 81π/2