Інтеграл/Приклади/Приклад 1

$$ L = \int_L dl

x = 2(cos t + t sin t)

y = 2(sin t - t cos t)

t \in [0;\frac{\pi}{2}]

L = \int dl = \int \sqrt{dx^2 + dy^2}

dx = (-2sint+2sin t+2tcost)dt = 2t cos t dt

dy = (2cost-2cost+2tsint)dt = 2t sint dt

\sqrt{dx^2+dy^2} = \sqrt{4t^2cos^2 t+4t^2 sin^2 t} dt = 2tdt

L = \int_{L} dl = \int_{0}^{\frac{\pi}{2}} 2t dt = t^2|_{0}^{\frac{\pi}{2}} = \frac{\pi^2}{4} $$