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Find the area of the region enclosed by one loop of the curve r=sin4Ɵ?

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.
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Dr. Juanita Wolff

Feb 20, 2025

sin4θ=0 4θ= 0, pi, 2pi, etcθ= 0, pi/4, pi/2, etcIntegrate (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... Show More
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Dr. Juanita Wolff

Feb 20, 2025

Calculating area using polar equation, between angles a and b:A = 1/2 ∫ₐᵇ r² dθNow curve r = sin(4θ) has 8 loops: http://www.wolframalpha.com/input/?i=polar+plot+r+...One loop is located between θ = 0 and θ = π/4A = 1/2 ∫[θ=0..π/4] sin²4θ dθA = 1/2 ∫[θ=0..π/4] 1/2 (1 - cos(8θ)) dθA = 1/4 ∫[θ=0..π/4] (1 - cos(8θ)) dθA = 1/4 (θ - 1/8 sin(8θ)) |[θ=0..π/4]A = 1/4 (π/4 - 1/8 sin(2π) - 0 + 1/8 sin(0))A = 1/4 (π/4)A = π/16... Show More
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