 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/13/b/solution

It is helpful first to separate an exact gradient:

F = \nabla (α x^{2} + α^{2} x y + αx z) + (0, β z, β y^{2}) .

2 α + α^{2}

for either path. On the curved path,

d y = 2 t d t

d z = 3 t^{2} d t

y = t^{2}

and

z = t^{3}

, so the remaining contribution is

β \int_{0}^{1} (z d y + y^{2} d z) = β \int_{0}^{1} (2 t^{4} + 3 t^{6}) d t = \frac{29 β}{35} .

Thus

\int F \cdot d r = α^{2} + 2 α + \frac{29 β}{35} .

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