Integrals and Volume?

Question

A cylindrical drill with radius 1 is used to bore a hole throught the center of a sphere of radius 2. Find the volume of the ring shaped solid that remains.

I know that you can't just subtract the volume of the cylinder, as that does not take into acount the rounded top and bottom. If you could help me set up this problem, I would really appreciate it.

Answer 1

First set up some equations for the sphere and the cylinder.

x^2 + y^2 = 4
If we rotate this about the x-axis we get a sphere of radius 2.
So let y = sqrt(4 - x^2). Rotating this about the x-axis gives the sphere of radius 2 still.

y = 1. Rotating this about the x-axis gives the cylinder of radius 1.

So if we look at the area bounded by the curves y = sqrt(4-x^2) and y=1, rotated about the x-axis, we get the volume we're looking for.

So the integral we're looking for is pi* Int ((sqrt(4-x^2))^2 - 1^2)dx (Using the disk method). And we can find our limits of integration by setting the two equations equal. sqrt(4-x^2)=1 when x = +- sqrt(3).

V = pi * Int from -sqrt(3) to sqrt(3) of 4-x^2 - 1 dx
V = pi * 3x - (1/3)x^3 Evaluated from -sqrt(3) to sqrt(3).
V = pi * 2 * (3sqrt(3) - (1/3)3^(3/2))
V = pi * 2 * (3sqrt(3) - sqrt(3))
V = 4 * pi * sqrt(3).

Answer 2

Use cylindrical shell method:
V = 2π ∫ rh dx
r = mean radius = x
h = height of partition = √(4 - x²)

Upper part of the solid
V1 = 2π ∫[1.. 2] x √(4 - x²) dx
u = (4 - x²)
du = -2x dx

V1 = -π ∫[3.. 0] u^½ du
= -π {(2/3) u^(3/2)} from 3 to 0
= π * 2/3 * 3^(3/2)
= 2π√3 ..... but this is just the upper part

∴ V = 4π√3.


d:

Note: if instead your V1 = 2π ∫[0.. 2] x √(4 - x²) dx then it becomes V1 = -π ∫[4.. 0] u^½ du = 2π/3 * 4^(3/2)
= 16π/3 which is the volume of the upper hemisphere.

Answer 3

I think before you take the integral you have to set the equation for the circle equal to x instead of y and then you subtract the 1 (do that befor integrating). Then continue normally.

Answer 4

Do you know the method of "cylindrical shells" for calculating the volumes of solids of revolution? I think that would be the easiest way to do this problem. See the website:

http://www.geocities.com/pkving4math2tor7/7_app_of_the_intgrl/7_03_02_finding_vol_by_using_cylind_shells.htm

Answer 5

You might want to do this in cylindrical coordinates. It gets messy in x-y-z coordinates.