A ball of radius 10 has a round hole of radius 5 drilled through its center. Find the volume of the resulting solid.

First you find the volume of the whole ball, then the volume of the round hole inside, then, you subtract the hole's volume from the whole ball's volume, leaving you with the volume of the resulting solid.

Perform a volume integration with the center of the sphere at the origin and the circular hole aligned with the z-axis. The integration will end at +/-a, where

a^2 = 10^2 - 5^2 = 75
a = 5 sqrt 3 = 8.6603
The volume will be twice the volume above the x-y plane.
V = 2*INTEGRAL pi (r^2 - 5^2) dz
(z = 0 to 8.6603)
r^2(z) = 100 - z^2

To find the volume of the resulting solid, we can break it down into two parts: the spherical cap and the cylindrical hole.

1. Spherical Cap:
The volume of a spherical cap can be calculated using the formula:
V_cap = (1/3)πh^2(3R - h)
where R is the radius of the sphere, and h is the height of the cap.
In this case, the radius of the sphere, R, is 10, and the height of the cap, h, can be found using the Pythagorean theorem:
h = √(R^2 - r^2)
where r is the radius of the cylindrical hole, which is 5 in this case.
So, plugging in the values, we have:
h = √(10^2 - 5^2) = √(100 - 25) = √75 ≈ 8.66
V_cap = (1/3)π(8.66)^2(3(10) - 8.66)

2. Cylindrical Hole:
The volume of a cylinder can be calculated using the formula:
V_cylinder = πr^2h
where r is the radius of the hole and h is the height of the hole.
In this case, the radius of the hole is 5, and the height of the hole is the same as the height of the spherical cap, which is 8.66.
So, plugging in the values, we have:
V_cylinder = π(5^2)(8.66)

Now, to find the volume of the resulting solid, we add the volume of the spherical cap and the volume of the cylindrical hole:
Volume = V_cap + V_cylinder

To find the volume of the resulting solid, we need to subtract the volume of the drilled hole from the volume of the ball.

1. Calculate the volume of the ball:
The volume of a ball is given by the formula V = (4/3) * π * r^3, where r is the radius.
Inserting the values into the formula, we get V_ball = (4/3) * π * (10^3) = (4/3) * π * 1000

2. Calculate the volume of the drilled hole:
The hole has a radius of 5, so we can use the same formula for the volume of a ball to calculate the volume of the drilled hole.
V_hole = (4/3) * π * (5^3) = (4/3) * π * 125

3. Subtract the volume of the drilled hole from the volume of the ball:
V_resulting_solid = V_ball - V_hole = ((4/3) * π * 1000) - ((4/3) * π * 125)

Simplifying the equation, we can factor out (4/3) * π:
V_resulting_solid = (4/3) * π * (1000 - 125)

Calculating the difference:
V_resulting_solid = (4/3) * π * 875

So, the volume of the resulting solid is approximately 2916.67 cubic units.