Volume of sphere?

Question

A ball of radius 12 has a round hole of radius 9 drilled through its center. Find the volume of the resulting solid.

Answer 1

let us fix our ball so that its center is at the origin...

the equation of the cross section of the ball in the first quadrant is x² + y² = 144

now, we are going to drill a horizontal hole... it becomes a band of height 9.

notice that there is a right angle with radius 12 and one leg 9.
the other leg is 3√7

In the first quadrant... the solid is from x = 0 to x = 3√7
the outer radius is the circle so that r1² = y² = 144 - x²
the inner radius is r2 = 9

(note... the integral is doubled because we need to consider first and second quadrants, but the integral is for the first quadrant initially)
V = 2π ∫[0.. 3√7] {144 - x² - 9²} dx
= 2π ∫[0.. 3√7] {63 - x²} dx

solve completely by this time...


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btw, the formula is 4/3 π b³ , where b² is the difference of the squares of the radii... the sphere's and the hole's.

Answer 2

(4/3)pi radius squared

and that's pi times radius squared