Esercizi Analisi matematica 2

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Calcolo del potenziale

Per avere il potenziale devo integrare sul cubo ∫_{x²+y²+z²≤R} ^-δG/_{√(x²+y²+(z-ξ)²)} ∫₀^R de ∫₀^2π dφ ∫₀^π dσ ^{-δG e² sinσ}/_{√(e² sin²σ + e² cos²σ - 2e cosσ ξ + ξ²)}

• X = e sinσ cosφ
• Y = e sinσ sinφ
• Z = e cosσ

∫₀^β ∫₀^2π ∫₀^π -δG e² sinθ ────────── = √e² - 2e ξ cosθ + ξ² = -δG ∫₀^R dε 2π e² (e² - 2e ξ cosθ + ξ²)^½2 / 2eξ|₀^π = -δG ∫₀^R 2π dε e² / eξ (e² + 2e ξ + ξ²)^½ - (e² - 2e ξ + ξ²)^½ = -2π δ/ξ G ∫₀^R dε e (|e + ξ| - |e - ξ|) = - M C_ξ / ξ

Calcolare il volume dell'insieme definito

V = { x ∈ ℝ³ : α + ax² + by² ≤ z ≤ β + cx + dy } α, a, b, c, d, β > 0, β > α

a ≠ b non è di notazione

(x, y, z) : z ≥ α + ax² + by²

D = C "(x, y)"

∫∫_D dxdy(β+Cx+dy-(a+ax²+by²)): ∫∫_D dxdy(Ψ₂(x,y)-Ψ₁(x,y))^o ≡ I

ξ=x²+y²-12 (∂z/∂x)²=2x=4x² (∂z/∂y)²=2y=4y² √(1+4x²+4y²)=√(1+4(x²+y²))

∫₀^2π ∫₂⁶ r√(1 + 4r²) dr dθ

sym;1 + 4r² = u du8r dr = du= ∫₀^7π dθsym; ∫₄⁸ 8r√(1 + 4r²) dv= ∫₀^7π dθsym; 1/8 ( (1 + 4v²)^{(1/3) + 1} / (1/3 + 1) )|₄⁸ - 1/8 . 2/3 (1 + 4v²)^3/2

Calcolare V.P.

∫₀^+∞ x^3/4 dx / ((x-1)(x+1)²)

x - 1 = 0 → x = 1

(x+1)² = x = -1

Polo ordine 2

Res (f(z), z=-1) = Res ( za=1   p=-3   q=1