 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021/6/solution

Parametrize the curve by

r (x) = (x, e^{x}, e^{x})

for

0 \leq x \leq 1

. Then

d r = (1, e^{x}, e^{x}) d x

and

F (r (x)) = (e^{2 x}, x, e^{x})

\int_{C} F \cdot d r = \int_{0}^{1} (2 e^{2 x} + x e^{x}) d x .

\int_{C} F \cdot d r = [e^{2 x} + (x - 1) e^{x}]_{0}^{1}, \int_{C} F \cdot d r = e^{2} .

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