Draw it carefully

Calculus Level 3

A body given by

{ ( x , y , z ) ∣ x 2 + y 2 + z 2 ≤ 2 , z ≥ 0 , x 2 + y 2 ≥ 1 } \left\{ {(x,y,z) | x^2+y^2+z^2 \leq 2 , z \geq 0 , x^2+y^2 \geq 1} \right\}

If the mass density of the body if given by

ρ ( x , y , z ) = z \rho(x,y,z)=z

so the body mass is

2 π 2\pi π 3 \frac{\pi}{3} π 4 \frac{\pi}{4} π \pi π 5 \frac{\pi}{5}

ابراهيم فقرا by ابراهيم فقرا , 23, Israel

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1 solution

Let's solve with cylindrical coordinates :- x = r c o s θ , y = r s i n θ , z = z x=rcos\theta,y=rsin\theta,z=z :-

∫ 0 1 ∫ 0 2 π ∫ 1 ( 2 − z 2 ) r z d r d θ d z = π 4 \int\limits_0^1\int\limits_0^{2\pi}\int\limits_1^{\sqrt{(2-z^2)}}rz \mathrm{d}r\mathrm{d}\theta\mathrm{d}z = \frac{\pi}{4}

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