 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-v/2/b/ii/solution

Compute the line integral on each straight segment. On the first,

y = z = 0

and only

d x

contributes; on the second,

x = 1, z = 0

and only

d y

contributes; on the third,

x = y = 1

and only

d z

contributes. Therefore

\int_{C} F \cdot d r = \int_{0}^{1} 3 x^{2} d x + \int_{0}^{1} 0 d y + \int_{0}^{1} 20 z^{2} d z = 1 + 0 + \frac{20}{3} = \frac{23}{3} .

Although

F_{x} = 9

on the second segment, that term does no work there because

d x = 0

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