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Triple integrals
- Intro Lesson: a17:06
- Intro Lesson: b14:39
- Intro Lesson: c10:08
- Lesson: 16:53
- Lesson: 211:06
- Lesson: 311:23
- Lesson: 422:35
- Lesson: 514:25
Triple integrals
Lessons
Triple Integrals with a Box Region
3 Cases of General Regions
The Use of Triple Integrals
If f(x,y,z) is continuous on a box region E=[a,b]×[c,d]×[e,f], then
∫∫∫Ef(x,y,z)dV=∫ef∫cd∫abf(x,y,z)dxdydz
3 Cases of General Regions
Suppose we are integrating f(x,y,z) on region E. There are 3 cases of finding region E.
Case 1: Region E is
(x,y)∈D
h1(x,y)≤z≤h2(x,y)
So,
V=∫∫∫Ef(x,y,z)dV=∫∫D[∫h1(x,y)h2(x,y)f(x,y,z)dz]dA
Case 2: Region E is
(y,z)∈D
h1(y,z)≤x≤h2(y,z)
So,
V=∫∫∫Ef(x,y,z)dV=∫∫D[∫h1(y,z)h2(y,z)f(x,y,z)dx]dA
Case 3: Region E is
(x,z)∈D
h1(x,z)≤y≤h2(x,z)
So,
V=∫∫∫Ef(x,y,z)dV=∫∫D[∫h1(x,z)h2(x,z)f(x,y,z)dy]dA
The Use of Triple Integrals
The volume of a 3D region E is given by the triple integral:
V=∫∫∫EdV
- IntroductionTriple Integrals Overview:a)
- Similar to rectangular regions, but 3D
- [a,b] is the x part
- [c,d] is the y part
- [e,f] is the z part
- Integral signs are interchangeable
- An Example
b)3 Cases of General Regions- Case 1: (x,y)∈D,h1(x,y)≤z≤h2(x,y)
- Case 2: (y,z)∈D,h1(y,z)≤x≤h2(y,z)
- Case 3: (x,z)∈D,h1(x,z)≤y≤h2(x,z)
- An Example
c)Use of Triple Integrals- Finds volume in 3D space
- An Example
- 1.Evaluate
∫02∫12∫−112xy2−zdzdydx - 2.Evaluate
∫0π∫0z∫0xsinxdydxdz - 3.Finding the Volume of 3D objects
Use triple integrals to determine the volume of the region below z=2, above z=1, bounded by y=x2−1 and y=1−x2 - 4.Evaluating Triple Integrals with Different types of Region E
Evaluate ∫∫∫ExdV where E is the region bounded by z=x2+y2−2 and the plane z=2. - 5.Evaluate ∫∫∫E2+ydV where E is the region below xy+3 above the region z=2, and bounded by 0≤x≤1,0≤y≤1