 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/1/c/solution

\nabla \cdot F = (2 y + 6 x^{5}) + (- 2 y + 4 y^{3}) + 1 = 6 x^{5} + 4 y^{3} + 1.

The volume is invariant under both

x \mapsto - x

and

y \mapsto - y

. Hence the integrals of the two odd terms vanish, and the flux is simply the volume. Using the circular cross-sections,

\iint_{S} F \cdot d S = π \int_{0}^{1} d z + π \int_{1}^{2} (2 - z) d z = \frac{3 π}{2} .

The seams between the surface pieces have zero surface measure and contribute no extra term.

 New to topics? Read the docs here!