Find the volume enclosed by the sphere x ^ 2 + y ^ 2 + z ^ 2 = R ^ 2 where R > 0 . (Hint: Use spherical coordinates)

Question

Find the volume enclosed by the sphere x ^ 2 + y ^ 2 + z ^ 2 = R ^ 2 where R > 0 . (Hint: Use spherical coordinates)

Answer 1

It is actually doable with cartesian coordinates:

At a given x coordinate, the cross section is a circle of radius √(R² - x²).

So you can integrate the disk area on x∈[-R,R].

V = ∫_[-R,R] π(R² - x²) = π(R²x - x³/3) |_[-R,R] = π(R³ - R³/3) - π(-R³ + R³/3) = 4πR³/3

You can use spherical coordinates as follows.

The Jacobian is r² sin θ.

V = ∫_[0,2π] ∫_[0,π] ∫_[0,R] r² sin θ dr dθ dφ

= ∫_[0,2π] ∫_[0,π] [r³ sin θ / 3]_[0,R] dθ dφ

= ∫_[0,2π] ∫_[0,π] R³ sin θ / 3 dθ dφ

= ∫_[0,2π] [-R³ cos θ / 3]_[0,π] dφ

= ∫_[0,2π] 2R³/3 dφ

= [2R³φ/3]_[0,2π]

= 4πR³/3