Calculate $int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)$












0












$begingroup$


Let $E_{3}:={ x in mathbb R^{3}: vert vert x vert vertleq 1}$



Calculate $int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)$



Using the polar coordinates formula:



$int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)=int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr$



Let $alpha=-cos{theta}$



Then:
$$int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr{=int_{0}^{1}int_{0}^{2pi}int_{theta=0}^{theta=pi}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}int_{-1}^{1}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}r^2[frac{1}{r}e^{ralpha}vert_{-1}^{1}]dvarphi dr\=int_{0}^{1}int_{0}^{2pi}re^{r}-re^{-r}dvarphi dr\=2piint_{0}^{1}re^r-re^{-r}dr\=2pi([e^r(r-1)vert_{0}^{1}-[-e^{-r}(r+1)vert_{0}^{1}])\=2pi[1+2e^{-1}-1]\=4pi e^{-1}}$$



I am unable to see where I went wrong...



Any help






real-analysis multivariable-calculus







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  • $begingroup$
    Your proof sounds perfectly fine to me.....
    $endgroup$
    – Mostafa Ayaz
    Jan 13 at 22:35
















0












$begingroup$


Let $E_{3}:={ x in mathbb R^{3}: vert vert x vert vertleq 1}$



Calculate $int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)$



Using the polar coordinates formula:



$int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)=int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr$



Let $alpha=-cos{theta}$



Then:
$$int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr{=int_{0}^{1}int_{0}^{2pi}int_{theta=0}^{theta=pi}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}int_{-1}^{1}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}r^2[frac{1}{r}e^{ralpha}vert_{-1}^{1}]dvarphi dr\=int_{0}^{1}int_{0}^{2pi}re^{r}-re^{-r}dvarphi dr\=2piint_{0}^{1}re^r-re^{-r}dr\=2pi([e^r(r-1)vert_{0}^{1}-[-e^{-r}(r+1)vert_{0}^{1}])\=2pi[1+2e^{-1}-1]\=4pi e^{-1}}$$



I am unable to see where I went wrong...



Any help






real-analysis multivariable-calculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your proof sounds perfectly fine to me.....
    $endgroup$
    – Mostafa Ayaz
    Jan 13 at 22:35














0












0








0





$begingroup$


Let $E_{3}:={ x in mathbb R^{3}: vert vert x vert vertleq 1}$



Calculate $int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)$



Using the polar coordinates formula:



$int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)=int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr$



Let $alpha=-cos{theta}$



Then:
$$int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr{=int_{0}^{1}int_{0}^{2pi}int_{theta=0}^{theta=pi}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}int_{-1}^{1}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}r^2[frac{1}{r}e^{ralpha}vert_{-1}^{1}]dvarphi dr\=int_{0}^{1}int_{0}^{2pi}re^{r}-re^{-r}dvarphi dr\=2piint_{0}^{1}re^r-re^{-r}dr\=2pi([e^r(r-1)vert_{0}^{1}-[-e^{-r}(r+1)vert_{0}^{1}])\=2pi[1+2e^{-1}-1]\=4pi e^{-1}}$$



I am unable to see where I went wrong...



Any help






real-analysis multivariable-calculus







share|cite|improve this question











$endgroup$




Let $E_{3}:={ x in mathbb R^{3}: vert vert x vert vertleq 1}$



Calculate $int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)$



Using the polar coordinates formula:



$int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)=int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr$



Let $alpha=-cos{theta}$



Then:
$$int_{0}^{1}int_{0}^{2pi}int_{0}^{pi}e^{-rcostheta}r^{2}sin{theta}dtheta dvarphi dr{=int_{0}^{1}int_{0}^{2pi}int_{theta=0}^{theta=pi}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}int_{-1}^{1}e^{ralpha}r^{2}dalpha dvarphi dr\=int_{0}^{1}int_{0}^{2pi}r^2[frac{1}{r}e^{ralpha}vert_{-1}^{1}]dvarphi dr\=int_{0}^{1}int_{0}^{2pi}re^{r}-re^{-r}dvarphi dr\=2piint_{0}^{1}re^r-re^{-r}dr\=2pi([e^r(r-1)vert_{0}^{1}-[-e^{-r}(r+1)vert_{0}^{1}])\=2pi[1+2e^{-1}-1]\=4pi e^{-1}}$$



I am unable to see where I went wrong...



Any help






real-analysis multivariable-calculus




real-analysis multivariable-calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 13 at 22:32









Mostafa Ayaz

17k31039




17k31039










asked Jan 13 at 21:08









MinaThumaMinaThuma

1968




1968












  • $begingroup$
    Your proof sounds perfectly fine to me.....
    $endgroup$
    – Mostafa Ayaz
    Jan 13 at 22:35


















  • $begingroup$
    Your proof sounds perfectly fine to me.....
    $endgroup$
    – Mostafa Ayaz
    Jan 13 at 22:35
















$begingroup$
Your proof sounds perfectly fine to me.....
$endgroup$
– Mostafa Ayaz
Jan 13 at 22:35




$begingroup$
Your proof sounds perfectly fine to me.....
$endgroup$
– Mostafa Ayaz
Jan 13 at 22:35










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