Question Number 179344 by neinhaltsieger369 last updated on 28/Oct/22
$$\: \\ $$$$\:\mathrm{Help}-\mathrm{me}! \\ $$$$\: \\ $$$$\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \int_{\mathrm{0}} ^{\:\mathrm{3}\boldsymbol{\mathrm{cos}}\:\boldsymbol{\phi}} \boldsymbol{\theta\mathrm{sin}}\:\boldsymbol{\phi\mathrm{d}\theta\mathrm{d}\phi} \\ $$$$\: \\ $$
Commented by CElcedricjunior last updated on 29/Oct/22
![k=β«_0 ^π β«_0 ^(3cosβ
) πsinβ
dπdβ
k=β«_0 ^π sinβ
[(1/2)π^2 ]_0 ^(3sinβ
) dβ
k=(9/2)β«_0 ^π sin^3 β
dβ
=(9/2)β«_0 ^π (sinβ
βsinβ
cos^2 β
)dβ
k=(9/2)[βcosβ
+((cos^3 β
)/3)]_0 ^π =(9/2)(1β(1/3)) k=3 dβ²ou^ β«_0 ^π β«_0 ^(3sinβ
) πsinβ
dπdβ
= 3 um ..............le celebre cedric junior........](https://www.tinkutara.com/question/Q179466.png)
$$\boldsymbol{{k}}=\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \int_{\mathrm{0}} ^{\mathrm{3}{cos}\boldsymbol{\emptyset}} \boldsymbol{\theta}{sin}\boldsymbol{\emptyset{d}\theta{d}\emptyset} \\ $$$$\boldsymbol{\mathrm{k}}=\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \boldsymbol{{sin}\emptyset}\left[\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\theta}^{\mathrm{2}} \right]_{\mathrm{0}} ^{\mathrm{3}\boldsymbol{{sin}\emptyset}} \boldsymbol{\mathrm{d}\emptyset} \\ $$$$\boldsymbol{\mathrm{k}}=\frac{\mathrm{9}}{\mathrm{2}}\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \boldsymbol{\mathrm{si}}\overset{\mathrm{3}} {\boldsymbol{\mathrm{n}}\emptyset\mathrm{d}\emptyset}=\frac{\mathrm{9}}{\mathrm{2}}\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \left(\boldsymbol{{sin}\emptyset}β\boldsymbol{{sin}\emptyset{co}}\overset{\mathrm{2}} {\boldsymbol{{s}}\emptyset}\right)\boldsymbol{{d}\emptyset} \\ $$$$\boldsymbol{{k}}=\frac{\mathrm{9}}{\mathrm{2}}\left[β\boldsymbol{\mathrm{cos}\emptyset}+\frac{\boldsymbol{\mathrm{co}}\overset{\mathrm{3}} {\boldsymbol{\mathrm{s}}\emptyset}}{\mathrm{3}}\right]_{\mathrm{0}} ^{\boldsymbol{\pi}} =\frac{\mathrm{9}}{\mathrm{2}}\left(\mathrm{1}β\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$\boldsymbol{\mathrm{k}}=\mathrm{3} \\ $$$$\boldsymbol{\mathrm{d}}'\boldsymbol{\mathrm{o}}\grave {\boldsymbol{\mathrm{u}}} \\ $$$$\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \int_{\mathrm{0}} ^{\mathrm{3}\boldsymbol{{sin}\emptyset}} \boldsymbol{\theta\mathrm{sin}\emptyset{d}\theta{d}\emptyset}=\:\mathrm{3}\:{um} \\ $$$$\: \\ $$$$…………..{le}\:{celebre}\:{cedric}\:{junior}…….. \\ $$$$ \\ $$
Commented by neinhaltsieger369 last updated on 01/Nov/22
$$\:\mathrm{Gracias} \\ $$
Answered by mr W last updated on 28/Oct/22
![=β«_0 ^Ο sin Ο(β«_0 ^(3 cos Ο) ΞΈdΞΈ)dΟ =β«_0 ^Ο sin Ο(((9 cos^2 Ο)/2))dΟ =β(9/2)β«_0 ^Ο cos^2 Οd(cos Ο) =β(9/2)[((cos^3 Ο)/3)]_0 ^Ο =(3/2)[1β(β1)] =3](https://www.tinkutara.com/question/Q179359.png)
$$=\int_{\mathrm{0}} ^{\pi} \mathrm{sin}\:\phi\left(\int_{\mathrm{0}} ^{\mathrm{3}\:\mathrm{cos}\:\phi} \theta{d}\theta\right){d}\phi \\ $$$$=\int_{\mathrm{0}} ^{\pi} \mathrm{sin}\:\phi\left(\frac{\mathrm{9}\:\mathrm{cos}^{\mathrm{2}} \:\phi}{\mathrm{2}}\right){d}\phi \\ $$$$=β\frac{\mathrm{9}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} \mathrm{cos}^{\mathrm{2}} \:\phi{d}\left(\mathrm{cos}\:\phi\right) \\ $$$$=β\frac{\mathrm{9}}{\mathrm{2}}\left[\frac{\mathrm{cos}^{\mathrm{3}} \:\phi}{\mathrm{3}}\right]_{\mathrm{0}} ^{\pi} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\left[\mathrm{1}β\left(β\mathrm{1}\right)\right] \\ $$$$=\mathrm{3} \\ $$
Commented by neinhaltsieger369 last updated on 29/Oct/22
$$\:\mathrm{Thank}\:\mathrm{you}! \\ $$