Question

Find the length of the cardioid r=5+5cosθ.

Answer 1

Hello,

r=5(1+cos t)

r'=5(-sin t)

r²+r'²= 25[(1+cos t)²+(-sin t)²]=50(1-cos t)=50 sin² (t/2)
Between 0 and π, sin x>0 ==>|sin x|=sin x
$l= 2*5* \int\limits^{\pi}_0{sin( \frac{t}{2} )} \, dt= 5[-cos (t/2)]_0^{\pi}\\\\ =5(0+1)=5$

Here is the method but i may have make some mistakes.