Triple integrals

Triple integrals

Lessons

Triple Integrals with a Box Region

If f(x,y,z)f(x,y,z) is continuous on a box region E=[a,b]×[c,d]×[e,f]E=[a,b] \times[c,d] \times [e,f] , then

Ef(x,y,z)dV=efcdabf(x,y,z)dxdydz \int \int \int_E f(x,y,z) dV = \int^f_e \int^d_c \int^b_a f(x,y,z)dxdydz


3 Cases of General Regions

Suppose we are integrating f(x,y,z)f(x,y,z) on region EE. There are 3 cases of finding region EE.

Case 1: Region EE is

(x,y)D(x,y) \in D
h1(x,y)zh2(x,y) h_1(x,y) \leq z \leq h_2(x,y)

So,

V=Ef(x,y,z)dV=D[h1(x,y)h2(x,y)f(x,y,z)dz]dAV = \int \int \int_E f(x,y,z)dV = \int \int_D [ \int^{h_2(x,y)}_{h_1(x,y)} f(x,y,z) dz]dA

Case 2: Region EE is

(y,z)D(y,z) \in D
h1(y,z)xh2(y,z) h_1 (y,z) \leq x \leq h_2(y,z)

So,

V=Ef(x,y,z)dV=D[h1(y,z)h2(y,z)f(x,y,z)dx]dAV = \int \int \int_E f(x,y,z)dV = \int \int_D [ \int^{h_2(y,z)}_{h_1(y,z)} f(x,y,z) dx]dA

Case 3: Region EE is

(x,z)D (x,z) \in D
h1(x,z)yh2(x,z) h_1 (x,z) \leq y \leq h_2 (x,z)

So,

V=Ef(x,y,z)dV=D[h1(x,z)h2(x,z)f(x,y,z)dy]dAV = \int \int \int_E f(x,y,z)dV = \int \int_D [ \int^{h_2(x,z)}_{h_1(x,z)} f(x,y,z) dy]dA


The Use of Triple Integrals

The volume of a 3D region EE is given by the triple integral:

V=EdVV = \int \int \int_E dV

  • Introduction
    Triple Integrals Overview:
    a)
    • Similar to rectangular regions, but 3D
    • [a,b][a,b] is the xx part
    • [c,d][c,d] is the yy part
    • [e,f][e,f] is the zz part
    • Integral signs are interchangeable
    • An Example

    b)
    3 Cases of General Regions
    • Case 1: (x,y)D,h1(x,y)zh2(x,y)(x,y) \in D,h_1(x,y) \leq z \leq h_2(x,y)
    • Case 2: (y,z)D,h1(y,z)xh2(y,z)(y,z)\in D, h_1(y,z) \leq x \leq h_2(y,z)
    • Case 3: (x,z)D,h1(x,z)yh2(x,z)(x,z)\in D, h_1(x,z) \leq y \leq h_2(x,z)
    • An Example

    c)
    Use of Triple Integrals
    • Finds volume in 3D space
    • An Example