Calculate $int_{E_{3}}e^{-z}dlambda^{3}(x,y,z)$
$begingroup$
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
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
$endgroup$
add a comment |
$begingroup$
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
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
$endgroup$
$begingroup$
Your proof sounds perfectly fine to me.....
$endgroup$
– Mostafa Ayaz
Jan 13 at 22:35
add a comment |
$begingroup$
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
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
$endgroup$
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
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
edited Jan 13 at 22:32
Mostafa Ayaz
17k31039
17k31039
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
asked Jan 13 at 21:08
MinaThumaMinaThuma
1968
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
Your proof sounds perfectly fine to me.....
$endgroup$
– Mostafa Ayaz
Jan 13 at 22:35