I = ∫ − 3 3 ∫ 0 9 − x 2 ∫ 0 9 − x 2 − y 2 x 2 + y 2 d z d y d x Converting to cylindrical coordinates. d z d y d x = ρ d z d ρ d θ x = ρ cos θ , y = ρ sin θ x 2 + y 2 = ρ 2 ( cos 2 θ + sin 2 θ ) = ρ 2 y = 0 , sin θ = 0 , θ = 2 π y = 9 − x 2 y = 9 − ρ 2 cos 2 θ x = − 3 , ρ cos θ = − 3 x = − 3 , ρ cos ( 2 π ) = ρ = − 3 x = 3 , ρ cos ( 2 π ) = ρ = 3 ∴ y = 3 sin θ = 9 − 9 cos 2 2 π = 9 − 9 = 0 , θ = 0 z = 9 − x 2 − y 2 = 9 − ρ 2 I = ∫ − 3 3 ∫ 0 9 − x 2 ∫ 0 9 − x 2 − y 2 x 2 + y 2 d z d y d x = ∫ − 3 3 ∫ 0 2 π ∫ 0 9 − ρ 2 ρ 2 ⋅ ρ d z d θ d ρ = ∫ − 3 3 ∫ 0 2 π ∫ 0 9 − ρ 2 ρ 2 d z d θ d ρ = ∫ − 3 3 ρ 2 d ρ ∫ 0 9 − ρ 2 d z ∫ 0 2 π d θ = 2 π ∫ − 3 3 ρ 2 d ρ ⋅ 1 ∣ 0 9 − ρ 2 = 2 π ∫ − 3 3 ρ 2 9 − ρ 2 d ρ Substitute ρ = 3 sin α ∴ I = 2 π ∫ − π 2 π 2 9 sin 2 α 9 − 9 sin 2 α ⋅ 3 cos α d α = 2 π ∫ 0 π 2 9 sin 2 α 9 − 9 sin 2 α ⋅ 3 cos α d α = 324 π ∫ 0 π 2 sin 2 α cos 2 α d α = 324 π ( α 8 + sin α cos α 8 − sin α cos 3 α 4 ) ∣ 0 π 2 = 324 ( π 16 ) = 81 π 2 4 Note: ∫ − a a f ( x ) d x = 2 ∫ 0 a f ( x ) d x , if f ( x ) is even. I = \int_{-3}^{3}\int_0^{\sqrt{9 - x^2}}\int_0^{\sqrt{9 - x^2 - y^2}} \sqrt{x^2 + y^2}\, \mathrm{d}z\, \mathrm{d}y\, \mathrm{d}x\\
\textsf{Converting to cylindrical coordinates.}\\
\, \mathrm{d}z\, \mathrm{d}y\, \mathrm{d}x = \rho\mathrm{d}z\, \mathrm{d}\rho\, \mathrm{d}\theta\\
x = \rho\cos{\theta}, y = \rho\sin{\theta}\\
x^2 + y^2 = \rho^2(\cos^2{\theta} + \sin^2{\theta}) = \rho^2\\
y = 0, \sin{\theta} = 0, \theta = 2\pi\\
y = \sqrt{9 - x^2}\\
y = \sqrt{9 - \rho^2\cos^2{\theta}}\\
x= -3, \rho\cos{\theta} = -3\\
x= -3, \rho\cos(2\pi) = \rho = -3\\
x= 3, \rho\cos(2\pi) = \rho = 3\\
\therefore y = 3\sin{\theta} = \sqrt{9 - 9\cos^2{2\pi}} = \sqrt{9 - 9} = 0, \theta = 0\\
z = \sqrt{9 - x^2 - y^2} = \sqrt{9 - \rho^2}\\
\begin{aligned}
I = \int_{-3}^{3}\int_0^{\sqrt{9 - x^2}}\int_0^{\sqrt{9 - x^2 - y^2}} \sqrt{x^2 + y^2}\, \mathrm{d}z\, \mathrm{d}y\, \mathrm{d}x &= \int_{-3}^{3}\int_0^{2\pi} \int_0^{\sqrt{9 - \rho^2}} \sqrt{\rho^2} \cdot \rho\mathrm{d}z\, \mathrm{d}\theta\, \mathrm{d}\rho
\\&=\int_{-3}^{3}\int_0^{2\pi} \int_0^{\sqrt{9 - \rho^2}} \rho^2\,\mathrm{d}z\, \mathrm{d}\theta\, \mathrm{d}\rho
\\&=\int_{-3}^{3}\rho^2\,\mathrm{d}\rho\int_0^{\sqrt{9 - \rho^2}} \mathrm{d}z \int_0^{2\pi}\, \mathrm{d}\theta
\\&= 2\pi\int_{-3}^{3}\rho^2\,\mathrm{d}\rho \cdot1 \vert_0^{\sqrt{9 - \rho^2}}
\\&= 2\pi\int_{-3}^{3}\rho^2\sqrt{9 - \rho^2}\,\mathrm{d}\rho
\end{aligned}\\
\textsf{Substitute}\, \rho = 3\sin\alpha\\
\begin{aligned}
\therefore I &= 2\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}9\sin^2{\alpha}\sqrt{9 - 9\sin^2\alpha}\cdot3\cos{\alpha}\mathrm{d}\alpha
\\&= 2\pi\int_{0}^{\frac{\pi}{2}}9\sin^2{\alpha}\sqrt{9 - 9\sin^2\alpha}\cdot3\cos{\alpha}\mathrm{d}\alpha
\\&= 324\pi\int_{0}^{\frac{\pi}{2}}\sin^2{\alpha}\cos^2{\alpha}\,\mathrm{d}\alpha
\\&= 324\pi\left(\frac{\alpha}{8} + \frac{\sin{\alpha}\cos{\alpha}}{8}- \frac{\sin{\alpha}\cos^{3}{\alpha}}{4}\right)\vert_{0}^{\frac{\pi}{2}}
\\&= 324\left(\frac{\pi}{16}\right) = \frac{81\pi^2}{4}
\end{aligned}\\
\textsf{Note:}\, \int_{-a}^{a} f(x)\, \mathrm{d}x = 2\int_{0}^{a} f(x)\, \mathrm{d}x, \,\,\textsf{if }\,f(x) \textsf{is even.} I = ∫ − 3 3 ∫ 0 9 − x 2 ∫ 0 9 − x 2 − y 2 x 2 + y 2 d z d y d x Converting to cylindrical coordinates. d z d y d x = ρ d z d ρ d θ x = ρ cos θ , y = ρ sin θ x 2 + y 2 = ρ 2 ( cos 2 θ + sin 2 θ ) = ρ 2 y = 0 , sin θ = 0 , θ = 2 π y = 9 − x 2 y = 9 − ρ 2 cos 2 θ x = − 3 , ρ cos θ = − 3 x = − 3 , ρ cos ( 2 π ) = ρ = − 3 x = 3 , ρ cos ( 2 π ) = ρ = 3 ∴ y = 3 sin θ = 9 − 9 cos 2 2 π = 9 − 9 = 0 , θ = 0 z = 9 − x 2 − y 2 = 9 − ρ 2 I = ∫ − 3 3 ∫ 0 9 − x 2 ∫ 0 9 − x 2 − y 2 x 2 + y 2 d z d y d x = ∫ − 3 3 ∫ 0 2 π ∫ 0 9 − ρ 2 ρ 2 ⋅ ρ d z d θ d ρ = ∫ − 3 3 ∫ 0 2 π ∫ 0 9 − ρ 2 ρ 2 d z d θ d ρ = ∫ − 3 3 ρ 2 d ρ ∫ 0 9 − ρ 2 d z ∫ 0 2 π d θ = 2 π ∫ − 3 3 ρ 2 d ρ ⋅ 1 ∣ 0 9 − ρ 2 = 2 π ∫ − 3 3 ρ 2 9 − ρ 2 d ρ Substitute ρ = 3 sin α ∴ I = 2 π ∫ − 2 π 2 π 9 sin 2 α 9 − 9 sin 2 α ⋅ 3 cos α d α = 2 π ∫ 0 2 π 9 sin 2 α 9 − 9 sin 2 α ⋅ 3 cos α d α = 324 π ∫ 0 2 π sin 2 α cos 2 α d α = 324 π ( 8 α + 8 sin α cos α − 4 sin α cos 3 α ) ∣ 0 2 π = 324 ( 16 π ) = 4 81 π 2 Note: ∫ − a a f ( x ) d x = 2 ∫ 0 a f ( x ) d x , if f ( x ) is even.
#339914 on Dec 2023