Krivuljni integral prve vrste na Geodetskom fakultetu
Na pismenom ispitu iz Vektorske analize 20. 6. 2011. je bio sljedeći zadatak:
Izračunajte
gdje je k prvi zavoj cilindrične zavojnice
Rješenje.
Izračunajmo prvo koliki je ds. Kako je
to je
![\begin{array}{rcl} ds &=& \displaystyle \sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}\, dt=\sqrt{(-a\sin t)^2+(a\cos t)^2+a^2}\, dt\\[15pt] &=& \displaystyle \sqrt{a^2\sin^2 t + a^2\cos^2 t + a^2}\, dt = a\sqrt{2}\, dt \end{array}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uZEzMkIv6qNGL042F-iysbw6f1e4ZcZjewkJTK1201TMlPB21lsmLq-lQpT2nWy4mtApGQU-NDpOEKtV104jXhAPNCSdz25qaQwhcMzWfJ3-aq-0XhcS6uYSklqDi0cP6CnDGRlQhi1r37H_3JzLTc6jP0jpNAerU5Ck2UogNjBOqEcsp8ufCDOu6GaDHklv5eiAIEPCcK-uYVsyuUkLDywBx8KnmD5fWqpzTZ1dwbGT4K7j-Nbwkh8DalMe6ixtfzYlhY-4g1UVfROj9MciPD5rlGIowe26kk_H9Js5UNMBWqjEaPQ9_ntfJBEvSLjCWzOugha5oCDtaCLVa7BXFqyHD86PEiWi_la9j-KeW10dqQcyvP69sEHxD7fXtSdXZqDaQmb8-dCF3NozOym8MOaQnOK9W08m5tW1S2Ki5TWWWgzay1GDoPAMooPCoFsGGoVEeko31Diw6SHrMos5J3kdanZXy342jEPIcHSbuQ31ZB9WK9vsCEuxmvvHV1tsKnOUNBGYufO9PXDQlg_kN6yVAopP--KQCH1x_nlQTMV_XpMcrWI6h0CkE2LGJatUdzokcbcNB5Da1XyrJptJCiN1jUsF8=s0-d "\begin{array}{rcl} ds &=& \displaystyle \sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}\, dt=\sqrt{(-a\sin t)^2+(a\cos t)^2+a^2}\, dt\\[15pt] &=& \displaystyle \sqrt{a^2\sin^2 t + a^2\cos^2 t + a^2}\, dt = a\sqrt{2}\, dt \end{array}")
pa je, jer su granice 
![\begin{array}{rcl} \displaystyle \int_k \frac{z^2}{x^2+y^2}\, ds &=& \displaystyle \int_0^{2\pi}\frac{(at)^2}{(a\cos t)^2 + (a\sin t)^2}\, a\sqrt{2}\, dt\\[20pt] &=& \displaystyle a\sqrt{2}\int_0^{2\pi} t^2\, dt = \frac{8\sqrt{2} \pi^3 a}{3} \end{array}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tECE-2-xk7z_kBSaxlRreQT5vRh29D53_yeQ0z3kcLALktVqatYDbArB3VWLAxU_zUWzCTBIEAw3U47oC8oYCwTBEhlHInv-nEhG9_x_49fzQnojxU3Mk7f6r0ns96-VlIzibbJfMPN-BPdagRVj9FIvo9cdqxHiFi7C2K2HdBdkg5t2jZujrp7qBRLyH17YJNoJF1VbMcgGv-kYzD6fCZhMGYrEE5v1GMZL6vi9RDTVn772x8xYZozgWvRYRt8eq7k2p4l1AkfoDQ2IqwII0ow4gTX7Qn4lr95uKkOOXEJFmO0iO_7Wa0Ceg7xkH6DUFSYWWl1hcR9GKz2z1vXcIfk4SvaBumSyjNYXVJFt1nwrecS-qOI85GbprL7o32xmyNnglm2vbTr2p-qQT8olaPQG84NUCETR97b-6etdrlu2lT8Ywmm4Jdn9HmfenfxKG8WIicXkkUtrgG9eSEmYiacwhUjM8IBNnAIMvM__N9TJtR3ur5LxQ91IJAK2Ya6HUBM5R5c3usLYCPsgYDzJWYPWBFg5U2bmVOcHRXYImlo71B8ktwC84YRDcNXzGLflkdPq5K-SEeHGTdo5o8w17GyJ6ZT50n7S1OX4yBGhjwq4G9Y6LKckU5Y0xdTbnVi0ZLeCgFmzzdXRKQKNKFN5NwkQ=s0-d "\begin{array}{rcl} \displaystyle \int_k \frac{z^2}{x^2+y^2}\, ds &=& \displaystyle \int_0^{2\pi}\frac{(at)^2}{(a\cos t)^2 + (a\sin t)^2}\, a\sqrt{2}\, dt\\[20pt] &=& \displaystyle a\sqrt{2}\int_0^{2\pi} t^2\, dt = \frac{8\sqrt{2} \pi^3 a}{3} \end{array}")
