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12922529
Cylindrical Coordinates
Descrição
Chapter 15.5 - 15.7 of MA 261, attempted to create one page for every note associated with Cylindrical Coordinates
Sem etiquetas
ma 261
calc 3
lesson 24
purdue
mathematics
Mapa Mental por
Mike Lone
, atualizado more than 1 year ago
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Criado por
Mike Lone
quase 7 anos atrás
17
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Resumo de Recurso
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 --->
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
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
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:
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
Anexos de mídia
Sketch The Solid By The Given Inequalites (binary/octet-stream)
Sketch The Solid By The Given Inequalites Answer (binary/octet-stream)
Evaluate ∭ √(X^2 + Y^2) D V, E Where E Is The Region That Lies Inside The Cylinder X^2 + Y^2 = 9 And Between (binary/octet-stream)
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