## Thomas' Calculus 13th Edition

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We have $\dfrac{\partial f_2}{\partial x}=8y \times \sin (y) \\ \dfrac{\partial f_1}{\partial y}=8x \times \cos (y)$ Now, $\oint_C F \cdot dr=\iint_S (\dfrac{\partial f_2}{\partial x}-\dfrac{\partial f_1}{\partial x}) dA$ $$\int_{0}^{\pi/2}\int_{0}^{\pi/2} (8y \times siny-8 \space x cos \space y) \space dy \space dx \\=\int_{0}^{\pi/2} (\pi^2 \sin x -8x) dx\\=(\pi^2 \cos x -4x^2)_{0}^{\pi/2} \\=0$$