Integral Calculus - Double Integral

Jumat, 10 Juli 2020

Double Integral

 

Functions of two variables: f(x, y), f(u, v), …

Double integrals:

Riemann sum:

Small changes: Δx_i, Δy_i

Regions of integration: R, S

Polar coordinates: r, θ

Area: A

Surface area: S

Volume of a solid: V

Mass of a lamina: m

Density: ρ(x, y)

First moments of inertia: I_x, I_y, I₀

Charge of a plate: Q

Charge density: σ(x, y)

Coordinates of center of mass:

Average of a function: μ

 

• Definition of Double Integral

  The double integral over a rectangle [a, b] × [c, d] 

  is defined to be

     

  Where (x_{i - 1}, x_i) × (y_{j- 1}, y_j), and Δx_i = x_i – x_{i – 1}, 

  Δy_j = y_j – y_{j – 1}.

 

Figure

 

     The double integral over a gereral region R is

          

     Where rectangle [a, b] × [c, d] contains R,

     g(x, y) = f(x, y) if f(x, y) is in R and g(x, y) = 0 

     otherwise.

 

Figure

 

Figure

 

 

Figure

Sumber

Labels: Mathematician

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