 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/2/c/solution

U (x, y) = x^{3} y + y^{4}

, since

\nabla U = (3 x^{2} y, x^{3} + 4 y^{3})

. The fundamental theorem for line integrals therefore reduces the line integral to its endpoints. The curve starts at

(0, 1)

and ends at

(2, 1)

, so

\int_{C} F \cdot d r = U (2, 1) - U (0, 1) = 9 - 1 = 8.

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