12922529
Cylindrical Coordinates
- 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 --->
- x = rcos(θ)
- 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
- Need to take given x, y,
z values and change
them into r, θ, z values,
that looks like -->
- Associated with webassign 24, Chapter 15.7 Triple Int. in Cylindrical Coord.
- 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
- θ = 3π/4
- 3. Solve for z
- z = z = 3
- Answer:
- (-3,3,3) = (√(18), 3π/4, 3)
- (-3,3,3) = (√(18), 3π/4, 3)
- z = z = 3
- θ = tan^-1(-1)
- θ = tan^-1(3/(-3))
- r = √(18)
- r = √((-3)^2 + 3^2)
- 1. Solve for r
- 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
- (-3,3,3)
- 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
-->
- End result example
-->
- 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
- −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.
- 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.
- 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
- 0 ≤ z ≤ 1
- z = 0 and z = 1
- x^2 + y^2 = r^2
- x^2 + y^2 = 9
- r^2 = 9
- r = 3
- 0 ≤ r ≤ 3
- 0 ≤ r ≤ 3
- r = 3
- r^2 = 9
- x^2 + y^2 = 9
- θ is not limited in this situation. The circle goes all the way around
- 0 ≤ θ ≤
2π
- 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:
- Solution:
- From the bounds, plug this directly into your integral, mutiply the function by "r" and solve (replacing
√(x^2 + y^2) with r)
- z = z
- Create bounds
- 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.
- Change from Rectangular to Cylindrical Coordinates. (Let r ≥ 0 and 0 ≤ θ ≤ 2π)
- 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
- ∫ [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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