Calcolare il volume dell’insieme

 

$A = \{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + 3 \le z \le 4 \sqrt{x^2 + y^2}\}$

 


Il volume dell’insieme $A$ vale

 

 

$\int \int \int_A dxdydz = \int \int_{B} (\int_{x^2 + y^2 + 3}^{4\sqrt{x^2 + y^2}} dz) dxdy$

 

con $B = \{(x,y) \in \mathbb{R}^2: x^2 + y^2 + 3 \le 4 \sqrt{x^2 + y^2}\}$.

 

$\int \int_{B} (\int_{x^2 + y^2 + 3}^{4\sqrt{x^2 + y^2}} dz) dxdy = \int \int_{B} (4 \sqrt{x^2 + y^2} – x^2 – y^2 – 3) dxdy$ (1)

 

Conviene passare in coordinate polari

 

$\{(x = \rho \cos(\theta)),(y = \rho \sin(\theta)):}$

 

$\rho \in [0, +\infty) \quad \quad \theta \in [0, 2 \pi]$ 

 

e la matrice Jacobiana vale

 

$J(\rho, \theta) = [(\frac{\partial}{\partial \rho} x, \frac{\partial}{\partial \theta} x),(\frac{\partial}{\partial \rho} y, \frac{\partial}{\partial \theta} y)] = [(\cos(\theta), – \rho \sin(\theta)),(\sin(\theta), \rho \cos(\theta))]$

 

Pertanto

 

$dxdy = |\det(J(\rho, \theta))|d \rho d \theta = |\rho \cos^2(\theta) + \rho \sin^2(\theta)|d \rho d \theta = |\rho|d \rho d \theta = \rho d \rho d \theta$

 

$x^2 + y^2 + 3 \le 4 \sqrt{x^2 + y^2} \quad \implies \quad \rho^2 + 3 \le 4 \rho \quad \implies \quad 1 \le \rho \le 3$

 

Quindi (1) equivale a

 

$\int_{0}^{2 \pi} \int_1^3 (4 \rho – \rho^2 – 3) \rho d \rho d \theta =  \int_{0}^{2 \pi} \int_1^3 (4 \rho^2 – \rho^3 – 3 \rho) d\rho d \theta = \int_{0}^{2 \pi} [\frac{4}{3} (\rho^3)_1^3 – \frac{1}{4} (\rho^4)_1^3 – \frac{3}{2} (\rho^2)_1^3]d \theta =$

$= 2 \pi (\frac{104}{3} – 20 – 12) = 2 \pi (\frac{104}{3} – 32) = 2 \pi (\frac{104 – 96}{3}) = \frac{16}{3} \pi$

 

FINE

 

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