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


  • 1.
    Evaluate

    0212112xy2zdzdydx \int_{0}^{2}\int_{1}^{2}\int_{-1}^{1} 2xy^{2} - zdzdydx

  • 2.
    Evaluate

    0π0z0xsinxdydxdz \int_{0}^{\pi}\int_{0}^{z}\int_{0}^{x} sin \, x dydxdz

  • 3.
    Finding the Volume of 3D objects
    Use triple integrals to determine the volume of the region below z=2 \, z = 2 \, , above z=1 \, z = 1\, , bounded by y=x21 \, y = x^{2} - 1 \, and y=1x2 \, y = 1 - x^{2}

  • 4.
    Evaluating Triple Integrals with Different types of Region E
    Evaluate ExdV \, \int\int\int_{E} \, xd V \, where E \, E \, is the region bounded by z=x2+y22 \, z = x^{2} + y^{2} -2 \, and the plane z=2 \, z = 2 .

  • 5.
    Evaluate E2+ydV \, \int\int\int_{E} \, 2 + ydV\, where E \, E \, is the region below xy+3 \, xy + 3 \, above the region z=2 \, z = 2 , and bounded by 0x1,0y1\, 0 \leq x \leq 1, 0 \leq y \leq 1