Wednesday, March 30, 2022

March Calculus Madness Sweet Sixteen - Day 15: r = α * (1 - cos Θ)

 ------------


Welcome to March Calculus Madness!


------------


r = α * (1 - cos Θ)


r = α * (1 - cos Θ)

r^2 = α^2 * (1 - 2 * cos Θ + cos^2 Θ)


dr/dΘ = α * sin Θ

(dr/dΘ)^2 = α^2 * sin^2 Θ



Area from 0 ≤ Θ ≤ 2*π


1/2 * ∫ α^2 * (1 - cos Θ)^2  dΘ from Θ = 0 to Θ = 2*π

= α^2/2 * ∫ 1 - 2 * cos Θ + cos^2 Θ dΘ from Θ = 0 to Θ = 2*π

= α^2/2 * ∫ 1 - 2 * cos Θ + 1/2 * cos(2*Θ) + 1/2 dΘ from Θ = 0 to Θ = 2*π

= α^2/2 * (3/2 * Θ - 2 * sin Θ + 1/4 * sin(2*Θ) from Θ = 0 to Θ = 2*π)

= 3/2 * π * α^2


Arc Length from 0 ≤ Θ ≤ 2*π


r^2 + (dr/dΘ)^2

= α^2 * (1 - 2 * cos Θ + cos^2 Θ) + α^2 * sin^2 Θ

= α^2 - 2  * α^2 * cos Θ + α^2 * (cos^2 Θ + sin^2 Θ)

= α^2 - 2  * α^2 * cos Θ + α^2 

= 2 * α^2 - 2 * α^2 * cos Θ

= 2 * α^2 *(1  - cos Θ)


Arc Length:

∫ 2 * α^2 *(1  - cos Θ) dΘ from Θ = 0 to Θ = 2*π

= α * √2 * ∫ √(1  cos Θ) dΘ from Θ = 0 to Θ = 2*π

= α * √2 * 4 * √2

= 8 * α


Eddie 


All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


BA-54: Real Estate Programs

BA-54: Real Estate Programs BA-54:  Chris won this calculator at HHC 2024 and donated it to me.   Much appreciation as always.  ...