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    Calculate the volume of a solid below the surface given by z = 16 xy + 200 and is above the region in the xy-plane bounded by y = x^2 and y = 8 – x^2 . (Answer: 12800/3)

    If you use the bounds 2,-2 in the thing below you get the correct answer but how come if you use the bound 2,0 and then multiply that answer by two you don't get the same volume? Isn't it symmetrical?


    [256x^{2}-32x^{4}+1600x-\frac{400x^{3}}{3}]

    Thanks
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    (Original post by CHEN20041)
    Calculate the volume of a solid below the surface given by z = 16 xy + 200 and is above the region in the xy-plane bounded by y = x^2 and y = 8 – x^2 . (Answer: 12800/3)

    If you use the bounds 2,-2 in the thing below you get the correct answer but how come if you use the bound 2,0 and then multiply that answer by two you don't get the same volume? Isn't it symmetrical?


    [256x^{2}-32x^{4}+1600x-\frac{400x^{3}}{3}]

    Thanks
    Your region of integration at the xy-plane might be but you're finding the volume of the solid, and this solid is not symmetrical about the x=0 plane, so you cannot do what you've tried to. If you were integrating z=16x^2y+200 (or any other even power) on the very same conditions, then this is symmetrical about x=0, hence you can apply your shortcut.
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    (Original post by RDKGames)
    Your region of integration at the xy-plane might be but you're finding the volume of the solid, and this solid is not symmetrical about the x=0 plane, so you cannot do what you've tried to. If you were integrating z=16x^2y+200 (or any other even power) on the very same conditions, then this is symmetrical about x=0, hence you can apply your shortcut.
    Okay I see... but how did u know that the volume in the question isn't symmetrical but you know z=16x^2y+200 is?

    I mean is there a quick way to know I don't study maths so don't know a lot about 3d geometry.
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    (Original post by CHEN20041)
    Okay I see... but how did u know that the volume in the question isn't symmetrical but you know z=16x^2y+200 is?

    I mean is there a quick way to know I don't study maths so don't know a lot about 3d geometry.
    Because in your case, I can choose some y, then I can plug in a negative number for x, and then it's positive version, and get two completely different answers. So, there is no symmetry in the x=0 plane.

    In my augmented example, if I plug in the same things, you'd notice that you'd get exactly the same answers whether you're on the positive side of the x-axis or the negative side. So that one is symmetrical about x=0 plane.


    Mathematically, your equation f(x,y)\neq f(-x,y) whereas my one has this property; f(-x,y)=16(-x)^2y+200=16(-1)^2(x)^2y+200=16x^2y+200=f(x,y)

    And note that I said any even power, so the property holds true if you you got an even power. As a rule of thumb, if you have z=f(x,y) and one (or both) of the variables in your expression are all raised to an even power, then the surface is symmetric in the plane x=0 or y=0. Having that, as long as the corresponding region of integration is also symmetric about 0, you can do your trick.
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    (Original post by RDKGames)
    Because in your case, I can choose some y, then I can plug in a negative number for x, and then it's positive version, and get two completely different answers. So, there is no symmetry in the x=0 plane.

    In my augmented example, if I plug in the same things, you'd notice that you'd get exactly the same answers whether you're on the positive side of the x-axis or the negative side. So that one is symmetrical about x=0 plane.


    Mathematically, your equation f(x,y)\neq f(-x,y) whereas my one has this property; f(-x,y)=16(-x)^2y+200=16(-1)^2(x)^2y+200=16x^2y+200=f(x,y)

    And note that I said any even power, so the property holds true if you you got an even power. As a rule of thumb, if you have z=f(x,y) and one (or both) of the variables in your expression are all raised to an even power, then the surface is symmetric in the plane x=0 or y=0.
    Ohh I see nice Thanks for the explanation
 
 
 
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