 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/7/ii/solution

On the specified curve

r (t) = (t, t, t^{2})

(1, 1, 2 t)

and

F (r (t)) = (μ t^{2}, t - λ t^{2}, t^{2} + (λ - 2 - μ) t) .

t + (λ - μ - 4) t^{2} + 2 t^{3}

. Hence

\int F \cdot d r = \int_{0}^{1} [t + (λ - μ - 4) t^{2} + 2 t^{3}] d t = \frac{λ - μ - 1}{3} .

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