 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017/2/solution

x = a u

y = b v

z = c w

. For nonzero

a, b, c

∣ ab c ∣

, and the transformed interior is the ball

u^{2} + v^{2} + w^{2} \leq 4

. Its radius is two, so

V = ∣ ab c ∣ \int_{0}^{2} \int_{0}^{π} \int_{0}^{2 π} ρ^{2} sin θ d ϕ d θ d ρ = \frac{32 π}{3} ∣ ab c ∣.

a, b, c

are taken as positive semiaxis parameters, this is

32 πab c /3

. The actual semiaxes are

2∣ a ∣, 2∣ b ∣, 2∣ c ∣

; omitting the factor two in the transformed radius would make the volume eight times too small.

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