determining content of a cross vault on a ceiling-verfication
I found this question as an old exam exercise, which I would like help for solving it properly...I've been starring at it for quiet a long time now.
As the title says, it is about determining the content of a cross vault on a ceiling.
I think it is defined by two half cylindric surfaces with radius 1
on a square $ [-1,1] times [-1,1] $ which are ''going through'' each other. I hope you understand what I mean ^^
The equations for the cylindric surfaces are given by
$ x_1^2 + x_3^2=1 , 0 leq x_3 , |x_2| leq 1 $
$x_2^2 + x_3 ^2 = 1 , 0leq x_3, |x_1| leq 1 $
Any help very appreciated
My progress on this task is following:
fix $x_3 $
therefore you get
$x_1^2= 1-x_3^2 $ and $ x_2^2=1-x_3^2$
so $ - sqrt{1-x_3^2} leq x_1, x_2 leq sqrt{1-x_3^2}$
the surface on the square, will come to:
$ 4(1-x_3^2) $
And
$ int_0^1 4(1-x_3^2) = 8/3 $
what do you think?
real-analysis multivariable-calculus proof-verification multiple-integral
asked Dec 3 '18 at 20:58
constant94
1396
add a comment |
I found this question as an old exam exercise, which I would like help for solving it properly...I've been starring at it for quiet a long time now.
As the title says, it is about determining the content of a cross vault on a ceiling.
I think it is defined by two half cylindric surfaces with radius 1
on a square $ [-1,1] times [-1,1] $ which are ''going through'' each other. I hope you understand what I mean ^^
The equations for the cylindric surfaces are given by
$ x_1^2 + x_3^2=1 , 0 leq x_3 , |x_2| leq 1 $
$x_2^2 + x_3 ^2 = 1 , 0leq x_3, |x_1| leq 1 $
Any help very appreciated
My progress on this task is following:
fix $x_3 $
therefore you get
$x_1^2= 1-x_3^2 $ and $ x_2^2=1-x_3^2$
so $ - sqrt{1-x_3^2} leq x_1, x_2 leq sqrt{1-x_3^2}$
the surface on the square, will come to:
$ 4(1-x_3^2) $
And
$ int_0^1 4(1-x_3^2) = 8/3 $
what do you think?
real-analysis multivariable-calculus proof-verification multiple-integral
asked Dec 3 '18 at 20:58
constant94
1396
add a comment |
I found this question as an old exam exercise, which I would like help for solving it properly...I've been starring at it for quiet a long time now.
As the title says, it is about determining the content of a cross vault on a ceiling.
I think it is defined by two half cylindric surfaces with radius 1
on a square $ [-1,1] times [-1,1] $ which are ''going through'' each other. I hope you understand what I mean ^^
The equations for the cylindric surfaces are given by
$ x_1^2 + x_3^2=1 , 0 leq x_3 , |x_2| leq 1 $
$x_2^2 + x_3 ^2 = 1 , 0leq x_3, |x_1| leq 1 $
Any help very appreciated
My progress on this task is following:
fix $x_3 $
therefore you get
$x_1^2= 1-x_3^2 $ and $ x_2^2=1-x_3^2$
so $ - sqrt{1-x_3^2} leq x_1, x_2 leq sqrt{1-x_3^2}$
the surface on the square, will come to:
$ 4(1-x_3^2) $
And
$ int_0^1 4(1-x_3^2) = 8/3 $
what do you think?
real-analysis multivariable-calculus proof-verification multiple-integral
asked Dec 3 '18 at 20:58
constant94
1396
I found this question as an old exam exercise, which I would like help for solving it properly...I've been starring at it for quiet a long time now.
As the title says, it is about determining the content of a cross vault on a ceiling.
I think it is defined by two half cylindric surfaces with radius 1
on a square $ [-1,1] times [-1,1] $ which are ''going through'' each other. I hope you understand what I mean ^^
The equations for the cylindric surfaces are given by
$ x_1^2 + x_3^2=1 , 0 leq x_3 , |x_2| leq 1 $
$x_2^2 + x_3 ^2 = 1 , 0leq x_3, |x_1| leq 1 $
Any help very appreciated
My progress on this task is following:
fix $x_3 $
therefore you get
$x_1^2= 1-x_3^2 $ and $ x_2^2=1-x_3^2$
so $ - sqrt{1-x_3^2} leq x_1, x_2 leq sqrt{1-x_3^2}$
the surface on the square, will come to:
$ 4(1-x_3^2) $
And
$ int_0^1 4(1-x_3^2) = 8/3 $
what do you think?
real-analysis multivariable-calculus proof-verification multiple-integral
real-analysis multivariable-calculus proof-verification multiple-integral
asked Dec 3 '18 at 20:58
constant94
1396
asked Dec 3 '18 at 20:58
constant94
1396
edited Dec 5 '18 at 19:31
asked Dec 3 '18 at 20:58
constant94
1396
asked Dec 3 '18 at 20:58
constant94
1396
asked Dec 3 '18 at 20:58
constant94
1396
1396
add a comment |
add a comment |
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