 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/7/a/iii/solution

Take the dot product of the given field with the upward vector area element. The surface relation simplifies its integrand:

F \cdot d S = [y (x + z) + x] d x d y = (1 - x^{3} + x) d x d y .

The projected domain is exactly the rectangle

1/2 \leq x, y \leq 1

, since

y > 0

determines a unique

z = (1 - x^{3}) / y - x

. Therefore the surface integral is

\int_{S_{1}} F \cdot d S = \frac{1}{2} \int_{1/2}^{1} (1 - x^{3} + x) d x = \frac{1}{2} [x - \frac{x ^{4}}{4} + \frac{x ^{2}}{2}]_{1/2}^{1} = \frac{41}{128} .

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