Find the exact length of the third side 40, 30

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sophiat17

29.06.2021

Mathematics

Call the path $\mathcal C$ . Then the work done by $\mathbf f(x,y,z)$ along $\mathcal C$ is given by the line integral, $\displaystyle\int_{\mathcal C}\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=2\pi}\mathbf f(x(t),y(t),z(t))\cdot\dfrac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt$
Swapping the $\mathbf{ijk}$ notation out for ordered component notation, I'll write
$\mathbf r(t)=(x(t),y(t),z(t))=(3\cos t,3\sin t,3t)$
so that
$\dfrac{\mathrm d\mathbf r}{\mathrm dt}=(-3\sin t,3\cos t,3)$
The line integral reduces to $\displaystyle\int_0^1(9\cos t,9\sin t,7)\cdot(-3\sin t,3\cos t,3)\,\mathrm dt$ $=\displaystyle\int_0^1(-27\cos t\sin t+27\sin t\cos t+21)\,\mathrm dt$ $=\displaystyle21\int_0^1\mathrm dt=21$

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