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.

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 ∫ₐᵇ r² dθ

    Now 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/4

    A = 1/2 ∫[θ=0..π/4] sin²4θ dθ

    A = 1/2 ∫[θ=0..π/4] 1/2 (1 - cos(8θ)) dθ

    A = 1/4 ∫[θ=0..π/4] (1 - cos(8θ)) dθ

    A = 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

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