What is distance around the graph of r = 12 sin θ ?

Question

What is distance around the graph of r = 12 sin θ ?

A	6π
B	36π
C	12π
D	144π

Answer 1

Verify that the curve completes one loop from 0 to π radians so that bounds the integral.

The arc length segment is given by √[(dy)² + (dx)²]

Since x and y are functions of θ via r, we determine each separately and determine the differentials.

r = 12 sinθ so x = 12 sin θ cos θ , y = 12 sin² θ

dx = 12 cos² θ - 12 sin² θ dθ = 12 cos (2θ) dθ, dy = 24 sin θ cos θ = 12 sin (2θ) dθ

Arc length L =_o∫^π √[(dy)² +(dx)² ] =_o∫^π √[(12 sin (2θ))² + (12 cos (2θ))² ]dθ =_o∫^π 12 dθ = 12π

Answer 2

r = 12sinθ; r² = 12rsinθ; x² + y² = 12y; x² + (y - 6)² = 36.

This is circle radius r = 6 and center at (0, 6).

So we have to find length of circle C = 2πr = 2π·6 = 12π

Answer is C