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Thread: triple integral - two questions

  1. #16
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA
    Let's do this. A general case you may find helpful. Tetrahedron-wise.

    The tetrahedron in the first octant bounded by the coordinate planes and TKH's plane: $\displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
    Of course, $\displaystyle a>0,b>0,c>0$

    6 different integrals and they all evaluate to $\displaystyle \frac{1}{6}abc$.

    I hope.

    $\displaystyle \int_{0}^{a}\int_{0}^{b(1-\frac{x}{a})}\int_{0}^{c(1-\frac{x}{a}-\frac{y}{b})}dzdydx$

    $\displaystyle \int_{0}^{c}\int_{0}^{a(1-\frac{z}{c})}\int_{0}^{b(1-\frac{x}{a}-\frac{z}{c})}dydxdz$

    $\displaystyle \int_{0}^{c}\int_{0}^{b(1-\frac{z}{c})}\int_{0}^{a(1-\frac{y}{b}-\frac{z}{c})}dxdydz$

    $\displaystyle \int_{0}^{b}\int_{0}^{a(1-\frac{y}{b})}\int_{0}^{c(1-\frac{x}{a}-\frac{y}{b})}dzdxdy$

    $\displaystyle \int_{0}^{a}\int_{0}^{c(1-\frac{x}{a})}\int_{0}^{b(1-\frac{x}{a}-\frac{z}{c})}dydzdx$

    $\displaystyle \int_{0}^{b}\int_{0}^{c(1-\frac{y}{b})}\int_{0}^{a(1-\frac{y}{b}-\frac{z}{c})}dxdzdy$


    Evaluate. They should all be (abc)/6.

    I think your second one is the same only arranged differently in the plane.
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  2. #17
    MHF Contributor
    Aug 2007
    Quote Originally Posted by kittycat View Post
    Why do we need to set x=0 in order to find the y limit? Why don't we set y=0 to find the x limit?
    To find the x, y limits in xy-plane, I usually use the projection of G on xy-plane ie. the region R.
    I see you don't have to use this method. Why?
    A very, very important part of my post was this, "(I just picked an order.)" When we did the first one, the other day, there were factors suggesting what might be an "easiest" way to go about it. In this case, ANY order will be just as simple or complicated as any other order.

    I was very careful NOT to say, "Set x = 0". What I said was, "In a world where x doesn't exist." (I think. I didn't look back to check.) This is only a reference to the nature of the triple integral. Once you have done the integration with respect to x, there is no more x. Put it out of your mind and go down one dimension. In the order I picked, it is now a 2D problem in y and z.

    This goes to the very heart of mathematics. Once you have considered something completely, why mess around considering it any more? In this case, we've already solved the x-part. Let it alone and move on.

    This also goes to teaching styles. If your instructor INSISTS that you do z first, I'm not very impressed. In my view, that would be leaning more toward rote memorization and less toward creative thinking. Personally, I prefer the latter. On the other hand, it absolutely would not hurt to do it multiple ways, just to gain experience. This IS what I always have done with volumes of revolution. If it's the "Shells" unit, sure I used the shells method, but I also did disks or washers, checked my methodology, and learned something from the exploration. Why not use the same idea of personal exploration here, and in every other unit you face in your career?!
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