 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-2/5/a/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Parametrize the straight line by

r (t) = (t, t, t)

0 \leq t \leq 1

, with

r^{'} (t) = (1, 1, 1)

. Substituting into the vector field and taking the dot product gives

v (r (t)) \cdot r^{'} (t) = (t + 1) e^{t} + 5 t^{4} .

Hence the direct line integral is

\int_{C} v \cdot d r = \int_{0}^{1} [(t + 1) e^{t} + 5 t^{4}] d t = [t e^{t} + t^{5}]_{0}^{1} = e + 1.

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