Integral Calculus - Triple Integral

Kamis, 23 Juli 2020

Triple Integral

 

Function of rhree variable: f(x, y, z), g(x, y, z), …

Triple integrals:

Riemann sum: 

Small hanges: Δx_i , Δy_j , Δz_k

Limits of integration: G, T, S

Cylindrical coordinates: r, θ, z

Spherical coordinates: r, θ, φ

Volume of a solid: V

Mass of a solid: m

Density: μ(x, y, z)

Coordinates of center of mass: 

First moments: M_xy, M_yz, M_xz

Moments of inertia: I_xy, I_yz, I_xz, I_x, I_y, I_z, I₀

• Definition of Triple Integral

   The triple integral over a parallelepiped 

   [a, b] × [c, d] × [r, s] is defined to be

    

  Where (u_i, v_j, w_k) is some point in the parallelepiped

  (x_{i –1} , x_i ) × (y_j–1 , y_i) × (z_k–1 , z_i), and Δx_i  = x_i – x_{i –1}, 

   Δy_j = y_j – y_{j –1} , Δz_k  = z_k – z_{k –1}.

• If f(x, y, z) ≥ 0 and G and T are nonoverlapping basic region, then

      

  Here G ∪ T is the union of the regions G and T.

 

• Evauation of Triple Integral by Repeated Integral

   If the solid G is the set of points (x, y, z) such that

   (x, y) ∈ R, χ₁ (x, y) ≤ z ≤ χ₂ (x, y), then

   Where R is profection of G onto the xy-plane.

      

   where R is projection of G onto the xy-plane.

 

  If the solid G is the set of points (x, y, z) such 

  that a ≤ x ≤ b, φ₁ (x) ≤ y ≤ φ₂ (x), 

  χ₁ (x, y) ≤ z ≤ χ₂ (x, y), then

  

• Triple Integral over Parallelepiped

   If G is a parallelepiped [a, b] × [c, d] × [r, s], then

   

   In the special case where the integrand f(x, y, z) can be 

   written as g(x) h(y) k(z) we have  

Sumber

Labels: Mathematician

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