this is an area and arc length in polar coordinates problem. What do i do with the 4? angles where
sin = 0 are at 0 and "pi". am i supposed to multiply the 4 with pi and zero ? im confused.
2 Answers

sin4θ=0
4θ= 0, pi, 2pi, etc
θ= 0, pi/4, pi/2, etc
Integrate (1/2)r^2dθ from θ= 0 to pi/4 to get one loop.
INT (1/2)(sin4θ)^2dθ for [0, pi/4]= .196, using my TI
🙂 R

Calculating area using polar equation, between angles a and b:
A = 1/2 â«ₐáµ rÂ² dÎ¸
Now curve r = sin(4Î¸) has 8 loops: http://www.wolframalpha.com/input/?i=polar+plot+r+...
One loop is located between Î¸ = 0 and Î¸ = Ï/4
A = 1/2 â«[Î¸=0..Ï/4] sinÂ²4Î¸ dÎ¸
A = 1/2 â«[Î¸=0..Ï/4] 1/2 (1  cos(8Î¸)) dÎ¸
A = 1/4 â«[Î¸=0..Ï/4] (1  cos(8Î¸)) dÎ¸
A = 1/4 (Î¸  1/8 sin(8Î¸)) [Î¸=0..Ï/4]
A = 1/4 (Ï/4  1/8 sin(2Ï)  0 + 1/8 sin(0))
A = 1/4 (Ï/4)
A = Ï/16