For the curve r = 2 sin 3(theta):
Find the arc length of one petal
ds = √(r^2 + r'^2)
r' = 6cos3θ
s = ∫[0,π/3] √(4sin^2(3θ) + 36cos^2(3θ)) dθ
= 2∫[0,π/3] √(1 + 8cos^2(3θ)) dθ
This is an elliptic integral. Good luck.
To find the arc length of one petal of the curve r = 2 sin(3θ), we can use the formula for finding the arc length of a polar curve. The general formula for arc length of a polar curve given by r = f(θ) is:
Arc length = ∫[a, b] sqrt(r^2 + (dr/dθ)^2) dθ
In this case, we need to find the arc length from one value of θ to the next value of θ where the curve completes one full petal.
To determine the limits of integration, we observe that for a full petal, the value of θ will vary from a value where r = 0 to the next value where r = 0 again. In this case, the curve completes one full petal when r = 0 at θ = 2π/3 and again at θ = 4π/3.
Now, let's differentiate the given equation r = 2sin(3θ) to find the expression for (dr/dθ):
(d/dr) r = (d/dr) 2sin(3θ)
1 = 6cos(3θ) (dθ/dθ)
1 = 6cos(3θ)
Now, substitute this expression into the arc length integral formula:
Arc length = ∫[2π/3, 4π/3] sqrt((2sin(3θ))^2 + (6cos(3θ))^2) dθ
Simplify the integral:
Arc length = ∫[2π/3, 4π/3] sqrt(4sin^2(3θ) + 36cos^2(3θ)) dθ
You can solve this integral numerically using integration techniques or technology, such as numerical methods or a software program like Mathematica, to find the exact value of the arc length of one petal.