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Triple integrals in spherical coordinates
- Intro Lesson: a3:03
- Intro Lesson: b1:54
- Intro Lesson: c18:03
- Lesson: 118:25
- Lesson: 214:23
- Lesson: 317:18
- Lesson: 417:22
Triple integrals in spherical coordinates
Lessons
Notes:
Spherical Coordinates
Equations to Convert From Cartesian to Spherical
Triple Integrals in Spherical Coordinates
Spherical Coordinates
There are times where instead of Cartesian coordinates, we use spherical coordinates for triple integrals. For spherical coordinates, instead of x,y,z's, we have ρ,θ,φ's. In other words,
(x,y,z)→(ρ,θ,φ)

Equations to Convert From Cartesian to Spherical
From the graph, we can obtain the following equations which will be useful for converting spherical to cartesian, or vice versa:
r=ρsinφ
z=ρcosφ
z2+r2=ρ2
Using the equations from past sections, we can also obtain more equations:
x=ρsinφcosθ
y=ρsinφsinθ
ρ2=x2+y2+z2
Triple Integrals in Spherical Coordinates
Suppose we want to convert a triple integral of f(x,y,z) in Cartesian Coordinates to spherical Coordinates on region E. Let region E be:
a≤ρ≤b
α≤θ≤β
δ≤φ≤γ
Then the conversion would be:
∫∫∫Ef(x,y,z)dV=∫δγ∫αβ∫abf(ρsinφcosθ,ρsinφsinθ,ρcosφ)ρ2sinφdpdθdφ
- IntroductionTriple Integrals in Spherical Coordinates Overview:a)Spherical Coordinates
- (x,y,z)→(ρ,θ,φ)
- Graph of the coordinates in 3D
b)Equations to Convert from Cartesian to Spherical- Pythagoras
- Trig Ratios
c)Triple Integrals in Spherical Coordinates- Region E
- Add an extra ρ2sinφ
- An Example
- 1.Convert the following integral into spherical coordinates
∫−2323∫−49−x249−x2∫3x2+3y29−x2−y2x3+xy2+xz2dzdydx - 2.Convert the following integral into spherical coordinates
∫−20∫−2−x20∫x2+y24−x2−y2x+y+dzdydx - 3.Evaluating Triple Integrals in Spherical Coordinates
Evaluate ∫∫∫E2x2+2y2dV where E is the region portion of x2+y2+z2=1 with z≥0. - 4.Evaluating Triple Integrals in Spherical Coordinates
Evaluate ∫∫∫E3dV where E is the region bounded by x2+y2+z2=4 and z=−x2+y2.