Free Printable Lemniskata

1.

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

a)

$r^2=8\cos2\varphi$

b)

$r^2=2\sqrt{2}\cos2\varphi$

c)

$r=8\cos2\varphi$

d)

$r=2\sqrt{2}\cos2\varphi$

2.

Nacrtajte lemniskatu čija polarna jednadžba glasi $r^2=8\cos2\varphi$

a)

b)

c)

d)

3.

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

a)

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

b)

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

c)

$\left(\pm2,0\right)$

d)

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

4.

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

a)

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

b)

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

c)

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

d)

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

5.

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

a)

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

b)

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

c)

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

d)

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

6.

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

a)

$\sqrt{3}+\frac{\pi}{3}$

b)

$\sqrt{3}-\frac{\pi}{3}$

c)

$1-\frac{\pi}{6}$

d)

$1+\frac{\pi}{6}$

Lemniskata

Total questions: 6

Worksheet time: 30mins

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