- #1
Mathitalian
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Homework Statement
Find the length of the curve:
[itex]\phi(t)=\left\{(5+\cos(3t))\cos(t), (5+\cos(3t))\sin(t) \right\}\mbox{ with } t\in [0, 2\pi][/itex]
Homework Equations
[itex]L_{\phi}= \int_{a}^{b}\sqrt{[x'(t)]^2+ [y'(t)]^2}\qquad (1.1)[/itex]
Where
[itex]x(t)= (5+\cos(3t))\cos(t)[/itex]
[itex]y(t)= (5+\cos(3t))\sin(t)[/itex]
[itex]a= 0\qquad b= 2\pi[/itex]
The Attempt at a Solution
Ok, i noticed that [itex]\phi(t)[/itex] is in this form:
[itex]\phi(t)=(r(t)\cos(t), r(t)\sin(t))[/itex]
so it can be expressed in polar form:
[itex]r= r(t)\iff r=5+\cos(3t) \quad t\in[0,2\pi][/itex]
so:
[itex]L_{\phi}=\int_{0}^{2\pi}\sqrt{[r(t)]^2+[r'(t)]^2}= [/itex]
[itex]= \int_{0}^{2\pi}\sqrt{(5+\cos(3t))^2+(-3\sin(3t))^2}dt[/itex]
but this integral is not so easy to solve :(
What can i do to solve it? I try to use wolfram but it gives me an approximate result.
[Sorry, my English is not so good, forgive me if there are mistakes]