- #1
Coolster7
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Homework Statement
Find the total length of the curve t --> (cos^3(t), sin^3(t)), and t is between 0 and ∏/2 where t is in radians. Find also
the partial arc length s(t) along the curve between 0 and ∏/2
Homework Equations
The length is given by:
S = ∫[itex]\sqrt{xdot^2 + ydot^2}[/itex] dt
The Attempt at a Solution
xdot = -3sin(t)cos^2(t) and ydot = 3sin^2(t)cos(t)
so subst. these into the formula gave me:
S = ∫[itex]\sqrt{9sin^2(t)cos^4(t)+9sin^4(t)cos^2(t)}[/itex]
I then used integration by subst. using u^2 = 9sin^2(t)cos^4(t)+9sin^4(t)cos^2(t)
The method I have been shown tells me to differentiate both sides and use this to subst. back into the intregal to solve.
After differentiation and some cancelling I have arrived at the following:
2u du = 18sin(t)cos^5(t) - 18cos(t)sin^5(t) dt
now I'm not sure what to do. Can anyone please help?