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Math Help - Volume of a pyramid

  1. #1
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    Volume of a pyramid

    I have two different problems that are built around the volume of the following:

    A three sided pyramid has sides of the following planes y= 0, y- x = 4, 2x + y + z = 4
    The base sits at the plane z = -6

    I have tried to draw a sketch but am still having trouble with this. I have the base being a platform at z = -6 and one side is along the y axis ( y = 0)

    I believe my function should be 4 - 2x - y and that I am dealing with a double integral.


    Would someone be willing to sketch this? Or help me with what my limits for x and y are?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Frostking View Post
    I have two different problems that are built around the volume of the following:

    A three sided pyramid has sides of the following planes y= 0, y- x = 4, 2x + y + z = 4
    The base sits at the plane z = -6

    I have tried to draw a sketch but am still having trouble with this. I have the base being a platform at z = -6 and one side is along the y axis ( y = 0)

    I believe my function should be 4 - 2x - y and that I am dealing with a double integral.


    Would someone be willing to sketch this? Or help me with what my limits for x and y are?
    Here's the sketch:



    Now, to find the limits of integration:

    If your dealing with a triple integral, we see that -6\leqslant z\leqslant 4-2x-y. Now, when y=0, y-x=4\implies -x=4\implies x=-4 and when y=0,z=-6, 2x+y+z=4\implies 2x-6=4\implies 2x=10\implies x=5. Thus, -4\leqslant x\leqslant 5. Finally, we can conclude that 0\leqslant y\leqslant x+4

    So your triple integral would be \int_{-4}^5\int_0^{x+4}\int_{-6}^{4-2x-y}\,dz\,dy\,dx

    The double integral would be \int_{-4}^5\int_0^{x+4}\left(10-2x-y\right)\,dy\,dx, which is a result from the triple integral.

    Does this make sense?
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  3. #3
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    Limits of integration

    Your sketch and detailed explanation helped a lot. I am still struggling with how to envision these shapes. Geometry and spacial problems are tough for me. Thanks very much for your time and effort. I will study your post and try to come up with the limits on my own. Frostking
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