Cylindrical Coordinates

Cylindrical Coordinates

1. Converting from SC to PC

   • x = rcos(θ)
   • y = rsin(θ)
   • z = z
   • 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

   • Need to take given x, y, z values and change them into r, θ, z values, that looks like -->
   • z = z
   • θ = tan^-1(y/x)
   • r = x^2 + y^2

3. Associated with webassign 24, Chapter 15.7 Triple Int. in Cylindrical Coord.

4. Examples

   • Change from Rectangular to Cylindrical Coordinates. (Let r ≥ 0 and 0 ≤ θ ≤ 2π)
     • (-3,3,3)
       • 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. Solve for r
           • r = √((-3)^2 + 3^2)
             • r = √(18)
             • 2. Solve for θ
               • θ = tan^-1(3/(-3))
                 • θ = tan^-1(-1)
                   • θ = 3π/4
                 • 3. Solve for z
                   • z = z = 3
                   • Answer:
                     • (-3,3,3) = (√(18), 3π/4, 3)
   • Sketch the solid described by the given inequalities
     • 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.
       • −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.
         • 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
           • End result example -->
   • Use cylindrical coordinates
     • 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.
       • Create bounds
         • z = z
           • z = 0 and z = 1
             • 0 ≤ z ≤ 1
         • x^2 + y^2 = r^2
           • x^2 + y^2 = 9
             • r^2 = 9
               • r = 3
                 • 0 ≤ r ≤ 3
         • θ is not limited in this situation. The circle goes all the way around
           • 0 ≤ θ ≤ 2π
         • Bounds are z from 0 to 1, r from 0 to 3, θ from 0 to 2π
           • From the bounds, plug this directly into your integral, mutiply the function by "r" and solve (replacing √(x^2 + y^2) with r)
             • Solution:

5. How a typical Triple intergral looks like

   • ∫ [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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