Question

Consider polar function r = 1 − cos(2θ), 0 ≤ θ ≤ 2π.

(a) Set up integral for, but do NOT evaluate, the arc length of the curve.
(b) Find the shaded area.

Answer 1

The formula for the arc length of a polar function is explained and an integral for the arc length of the given function is set up. The formula for finding the shaded area of a polar function is explained and an integral for the shaded area of the given function is set up.

Step-by-step explanation:

Polar Function and Arc Length

The given polar function is r = 1 - cos(2θ), 0 ≤ θ ≤ 2π.

(a) Arc Length

To find the arc length, we can use the formula:
Z = ∫√(r² + (dr/dθ)²) dθ

But we do not evaluate this integral.

(b) Shaded Area

Using the given polar function:
A = (1/2) ∫(1 - cos(2θ))² dθ

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