Math Problem Statement

The evaluation of
∫ ∫ D 4 x y ( x 2 + y 2 + 4 ) 3 d A , where
D is the entire first quadrant, can be accomplished by :

lim b → ∞

lim d → ∞

∫ b 0 ∫ d 0 4 x y ( x 2 + y 2 + 4 ) 3 d x d y

Work through the appropriate substitutions to complete the integrals, and evaluate the limits. What is the value?

Solution

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Double Integrals
Polar Coordinates
Substitution in Integrals

Formulas

Double integral ∫∫ f(x, y) dA
Polar coordinates: x = r cos(θ), y = r sin(θ)
Integral transformation: dx dy = r dr dθ
Trigonometric identity: cos(θ)sin(θ) = 1/2 sin(2θ)
Substitution: u = r^2 + 4

Theorems

Fubini's Theorem (for evaluating double integrals)
Substitution rule for definite integrals

Suitable Grade Level

Undergraduate (Calculus II or Multivariable Calculus level)