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Gabry89
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I don't understand what's wrong with it:
Gabry89 said:^i think I've understood what you mean but i just don't realize where's the case x+z > 1, i just see x + z <= 1
Gabry89 said:… i just don't realize where's the case x+z > 1, i just see x + z <= 1
Gabry89 said:^yes, that's not what i mean, i mean that
0<= y <= 1-x-z then x+z <= 1 not x+z > 1 as the case that tiny-tim was talkin' about
Char. Limit said:What tiny-tim is talking about is that if you want to integrate over the entire circle, you have to then subtract the part of the circle where x+z>1.
Gabry89 said:thanks, Happy new year too ;)
I thought about that, here's what I've done:
Looks good to me. Do the integration & see if the answer checks out.Gabry89 said:^like this then:
SammyS said:Looks good to me. Do the integration & see if the answer checks out.
If you are allowed to do part of the integration in rectangular coordinates, then using rectangular coordinates for [tex]\displaystyle 0\le\varphi \le\pi/2[/tex] appears to be straight forward.Gabry89 said:Ok, even if the first one looks kinda complicated 'cause i get (talkin' about [tex]\rho)[/tex] [tex]\int[/tex] from 0 to 1/(sin[tex]\varphi[/tex]+cos[tex]\varphi[/tex]) of [tex]\rho[/tex]^2 and besides the same for [tex]\rho[/tex]^3, then is there an easier way to solve these integrals than e.g. calculating the cube?
the second one gives me a part of the answer so it seems right.
SammyS said:
By the way, what part of the answer have you gotten thus far? ... What part of the integral have you done?
The above identities don't really help.SammyS said:
It may be helpful to know that [tex] \sin x+\cos x=\sqrt{2}\,\sin\left(x+{{\pi}\over{4}}\right)= \sqrt{2}\,\cos\left(x-{{\pi}\over{4}}\right)\,.[/tex]
So that [tex] {{1}\over{\sin \varphi+\cos \varphi}}=\sqrt{2}\,\csc\left(\varphi+{{\pi}\over{4}}\right)= \sqrt{2}\,\sec\left(\varphi-{{\pi}\over{4}}\right)\,.[/tex]
Gabry89 said:i did [tex]\int_{\pi/2}^{2\pi}[/tex] [tex]\int_0^1[/tex] [tex]\int_0^{1-\rho\cos\varphi-\rho\sin\varphi}\rho^2\,\cos\varphi\ dy\ d\rho\ d\varphi[/tex]
which gives me -1/3 + 1/8 -(3pi)/16 so a part of the right result ( we can see -(3pi)/16)
Gabry89 said:^like this then:
A triple integral in the cylindrical coordinate system is a mathematical tool used to calculate the volume of a three-dimensional shape in cylindrical coordinates. It involves integrating a function over a region in the cylindrical coordinate system.
The steps to calculate a triple integral using the cylindrical coordinate system are:
The cylindrical coordinate system is useful for calculating triple integrals when the shape being integrated over has cylindrical symmetry, meaning it can be described using a cylinder or circular shape. It is also useful when the limits of integration are easier to define in cylindrical coordinates.
Yes, a triple integral using the cylindrical coordinate system can be converted to rectangular coordinates. This can be done by using a change of variables formula and adjusting the limits of integration accordingly.
Some real-life applications of calculating triple integrals using the cylindrical coordinate system include calculating the volume of cylindrical objects such as pipes or tanks, determining the mass or density of objects with cylindrical symmetry, and calculating electric or gravitational fields in cylindrical systems.