Lemniskata

Lemniskata

University

6 Qs

quiz-placeholder

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Lemniskata

Lemniskata

Assessment

Quiz

Mathematics

University

Practice Problem

Medium

Created by

Matija Basic

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6 questions

Show all answers

1.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Napišite polarnu jednadžbu lemniskate čija jednadžba u Kartezijevim koordinatama glasi (x2+y2)2=8(x2y2)\left(x^2+y^2\right)^2=8\left(x^2-y^2\right)   

 r2=8cos2φr^2=8\cos2\varphi  

 r2=22cos2φr^2=2\sqrt{2}\cos2\varphi  

 r=8cos2φr=8\cos2\varphi  

 r=22cos2φr=2\sqrt{2}\cos2\varphi  

2.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Nacrtajte lemniskatu čija polarna jednadžba glasi r2=8cos2φr^2=8\cos2\varphi   

Media Image
Media Image
Media Image
Media Image

3.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Odredite presjek kružnice x2+y2=4x^2+y^2=4   i lemniskate  (x2+y2)2=8(x2y2).\left(x^2+y^2\right)^2=8\left(x^2-y^2\right). 

 (±2, ±2)\left(\pm\sqrt{2},\ \pm\sqrt{2}\right)  

 (±1, ±3)\left(\pm1,\ \pm\sqrt{3}\right)  

 (±2,0)\left(\pm2,0\right)  

 (±3, ±1)\left(\pm\sqrt{3},\ \pm1\right)  

4.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Odredite granice integracije za kut pri računanju površine dijela ravnine izvan kružnice  x2+y2=4x^2+y^2=4 , a unutar lemniskate  (x2+y2)2=8(x2y2)\left(x^2+y^2\right)^2=8\left(x^2-y^2\right) u desnoj poluravnini.

 φ[π2, π2]\varphi\in\left[-\frac{\pi}{2},\ \frac{\pi}{2}\right]  

 φ[π6, π6]\varphi\in\left[-\frac{\pi}{6},\ \frac{\pi}{6}\right]  

 φ[π3, π3]\varphi\in\left[-\frac{\pi}{3},\ \frac{\pi}{3}\right]  

 φ[π4, π4]\varphi\in\left[-\frac{\pi}{4},\ \frac{\pi}{4}\right]  

5.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Postavite integral kojim se računa površina dijela ravnine izvan kružnice x2+y2=4x^2+y^2=4 , a unutar lemniskate  (x2+y2)2=8(x2y2).\left(x^2+y^2\right)^2=8\left(x^2-y^2\right). Uzmite u obzir lijevu i desnu poluravninu!

 I=2π6π648cos2φr dr dφI=2\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\int_4^{8\cos2\varphi}r\ dr\ d\varphi  

 I=2π6π6222cos2φ dr dφI=2\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\int_2^{2\sqrt{2}\cos2\varphi}\ dr\ d\varphi  

 I=40π6222cos2φr dr dφI=4\int_0^{\frac{\pi}{6}}\int_2^{2\sqrt{2\cos2\varphi}}r\ dr\ d\varphi  

 I=40π6222cos2φr dr dφI=4\int_0^{\frac{\pi}{6}}\int_2^{2\sqrt{2}\cos2\varphi}r\ dr\ d\varphi  

6.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

 Izračunajte integral I=0π6222cos2φr dr dφ.I=\int_0^{\frac{\pi}{6}}\int_2^{2\sqrt{2\cos2\varphi}}r\ dr\ d\varphi. 

 3+π3\sqrt{3}+\frac{\pi}{3}  

 3π3\sqrt{3}-\frac{\pi}{3}  

 1π61-\frac{\pi}{6}  

 1+π61+\frac{\pi}{6}  

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