Cylindrical Coordinates

Mike Lone
Mind Map by Mike Lone, updated more than 1 year ago
Mike Lone
Created by Mike Lone almost 4 years ago
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Chapter 15.5 - 15.7 of MA 261, attempted to create one page for every note associated with Cylindrical Coordinates

Resource summary

Cylindrical Coordinates
  1. Converting from SC to PC
    1. x = rcos(θ)
      1. y = rsin(θ)
        1. z = z
          1. Need to take given r, θ, z values and change them into x,y, and z values, that looks like --->
          2. Converting from PC to SC
            1. Need to take given x, y, z values and change them into r, θ, z values, that looks like -->
              1. z = z
                1. θ = tan^-1(y/x)
                  1. r = x^2 + y^2
                  2. Associated with webassign 24, Chapter 15.7 Triple Int. in Cylindrical Coord.
                    1. Examples
                      1. Change from Rectangular to Cylindrical Coordinates. (Let r ≥ 0 and 0 ≤ θ ≤ 2π)
                        1. (-3,3,3)
                          1. To do this, remember the conversions needed for cylindrical coordinate format (r,θ,z): 1. r = √(x^2+y^2) 2. θ = tan^-1(y/x) 3. z = z
                            1. 1. Solve for r
                              1. r = √((-3)^2 + 3^2)
                                1. r = √(18)
                                  1. 2. Solve for θ
                                    1. θ = tan^-1(3/(-3))
                                      1. θ = tan^-1(-1)
                                        1. θ = 3π/4
                                        2. 3. Solve for z
                                          1. z = z = 3
                                            1. Answer:
                                              1. (-3,3,3) = (√(18), 3π/4, 3)
                              2. Sketch the solid described by the given inequalities
                                1. I like to look at the z first, z is still going to be z. So make sure that they all have the same z.
                                  1. −π/2 ≤ θ ≤ π/2 means that it is a half circle. So any shape that doesn't look like a half circle is wrong.
                                    1. 0 ≤ r ≤ 3 means that the radius can be a maximum of 3, but it comes down to zero. Meaning it contains a parabola of some sort
                                      1. End result example -->
                                2. Use cylindrical coordinates
                                  1. Evaluate ∭ √(x^2 + y^2) dV, E where E is the region that lies inside the cylinder x^2 + y^2 = 9 and between the planes z = 0 and z = 1.
                                    1. Create bounds
                                      1. z = z
                                        1. z = 0 and z = 1
                                          1. 0 ≤ z ≤ 1
                                        2. x^2 + y^2 = r^2
                                          1. x^2 + y^2 = 9
                                            1. r^2 = 9
                                              1. r = 3
                                                1. 0 ≤ r ≤ 3
                                          2. θ is not limited in this situation. The circle goes all the way around
                                            1. 0 ≤ θ ≤ 2π
                                            2. Bounds are z from 0 to 1, r from 0 to 3, θ from 0 to 2π
                                              1. From the bounds, plug this directly into your integral, mutiply the function by "r" and solve (replacing √(x^2 + y^2) with r)
                                                1. Solution:
                                      2. How a typical Triple intergral looks like
                                        1. ∫ [from "z" final to "z" initial] ∫ [from "θ" final to "θ" initial] ∫ [from "r" final to "r" initial] {(f(r,θ, z) (r))} drdθdz
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