z=arcsin(1/5)=0.201358 is the angle about (0,0) where the circles intersect. So, the arc length on the outer circle is indeed
∫[z,π-z] 5 dθ = 13.6944
But now you have to add the arc on the inner circle to complete the perimeter
∫[z,π-z] 1 dθ = 2.73888
so, the whole perimeter is 16.4333
You can also do it without calculus, knowing that s=rθ, and using the same angles. But you have to note that the subtended angle in the larger circle is twice the angle about (0,0).
Find the length of the entire perimeter of the region inside r=5sin(theta) but outside r=1.
1=5sin(theta)
theta=arcsin(1/5)
r'=5cos(theta)
I tried the integral between arcsin(1/5) and pi-arcsin(1/5) of (((5sin(theta))^2+(5cos(theta))^2))^1/2 which gives me 13.694 Webworks (a math homework website) says that my answer is wrong.
1 answer