Question

Let c be parametrized by x = et sin (10t) and y = et cos (10t) for 0 ≤ t ≤ 2. find the length l of

Answer 1

$C:x(t)=e^t\sin10t;\,y(t)=e^t\cos10t;\,0\le t\le2$

$\displaystyle\int_0^2\sqrt{x'(t)^2+y'(t)^2}\,\mathrm dt$
$\displaystyle=\int_0^2\sqrt{(e^t\sin10t+10e^t\cos10t)^2+(e^t\cos10t-10e^t\sin10t)^2}\,\mathrm dt$
$\displaystyle=\int_0^2e^t\sqrt{\sin^210t+20\sin10t\cos10t+100\cos^210t+\cos^210t-20\sin10t\cos10t+100\sin^210t}\,\mathrm dt$
$=\displaystyle\int_0^2e^t\sqrt{1+100}\,\mathrm dt$
$=\displaystyle\sqrt{101}(e^2-1)$