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Triple integrals

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Intros
Lessons
  1. Triple Integrals Overview:
    • 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
  3. 3 Cases of General Regions
    • Case 1: (x,y)∈D,h1(x,y)≤z≤h2(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)≤x≤h2(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)≤y≤h2(x,z)(x,z)\in D, h_1(x,z) \leq y \leq h_2(x,z)
    • An Example
  4. Use of Triple Integrals
    • Finds volume in 3D space
    • An Example
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Examples
Lessons
  1. Evaluate

    ∫02∫12∫−112xy2−zdzdydx \int_{0}^{2}\int_{1}^{2}\int_{-1}^{1} 2xy^{2} - zdzdydx
    1. Evaluate

      ∫0π∫0z∫0xsin xdydxdz \int_{0}^{\pi}\int_{0}^{z}\int_{0}^{x} sin \, x dydxdz
      1. 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=x2−1  \, y = x^{2} - 1 \, and  y=1−x2 \, y = 1 - x^{2}
        1. Evaluating Triple Integrals with Different types of Region E
          Evaluate  ∫∫∫E xdV  \, \int\int\int_{E} \, xd V \, where  E  \, E \, is the region bounded by  z=x2+y2−2  \, z = x^{2} + y^{2} -2 \, and the plane  z=2 \, z = 2 .
          1. Evaluate  ∫∫∫E 2+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  0≤x≤1,0≤y≤1\, 0 \leq x \leq 1, 0 \leq y \leq 1
            Topic Notes
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            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=∫ef∫cd∫abf(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)≤z≤h2(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)≤x≤h2(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)≤y≤h2(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