volume of the solid formed by the revolving region

Question

volume of the solid formed by the revolving region bounded by:

y=√4-x², x=0, y=0 about the y axis

Answer 1

Sob, since you are in calculus, my guess is that you need to find this volume by integration.

First, start by drawing a picture of a semi-circle from x = -2 to x = 2 with a max height of y = 2.

Now, draw in that semi-circle a horizontal disk (r = x) and ht = dy.

dV = (A of disk) dy

-         Note: A of disk = π r² = π x²  since r = x

V = ∫ (from y = 0 to 2) of A dy

= ∫ (0 to 2) of π x² dy

= ∫ (0 to 2) of π (4 – y²) dy                 (solve orig equation for x² = 4 – y² & substitute in)

= π [4y – (y³/3) ] 0 to 2

= π [ (4 x 2) – (2³)/3 ]

= π [ 8 – 8/3]

= π [24/3 – 8/3]

= 16/3 π

Of course, if your teacher allows you to find the volume with simple geometry, you would just find the volume of the hemisphere described, which = 1/2 x 4/3 π r³ = 16/3 π

Answer 2

This would be half of a sphere of radius 2, so the volume is 1/2 * 4/3 * 2³ π= 16/3 π