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Math Help - Riemann Sum Help

  1. #1
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    Exclamation Riemann Sum Help

    Can somone find the Riemann Sum of this intergral and also draw the volume of the solid:

    \int_0^1(-x+1)dx


    Thanx
    Last edited by Nimmy; May 28th 2006 at 06:08 AM.
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  2. #2
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    Since, f(x)=-x+1 is countinous on [0,1] its is integrable-meaning its Riemann Sum exists. Since the Riemann sum if its exists is independent of the partion of the interval (mathematical term is well-defined) we can use any one. The one I am going to use is the right-endpoint partion which is,
    \lim_{n\to\infty}\sum^n_{k=1}f(a+k\Delta x)\Delta x
    Where,
    a=0 and \Delta x=\frac{b-a}{n}=\frac{1-0}{n} and f(x)=-x+1 thus what we have is,
    \lim_{n\to\infty}\sum^n_{k=1}\left(1-\frac{k}{n}\right)\frac{1}{n}= \lim_{n\to\infty}\sum^n_{k=1}\frac{1}{n}-\frac{k}{n^2}
    Using the rule,
    \sum_{k=1}^n k=\frac{n(n+1)}{2}
    we have,
    \lim_{n\to\infty}1-\frac{n(n+1)}{2n^2}=1-1/2=1/2
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  3. #3
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    Can someone draw the solid of the figure?
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  4. #4
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    Quote Originally Posted by Nimmy
    Can someone draw the solid of the figure?
    Not sure what you mean?
    It not a solid.
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  5. #5
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    Quote Originally Posted by Nimmy
    Can someone draw the solid of the figure?
    Hello,

    are you looking for a cone?
    The line y = -x+1 has the x-axis as rotation axis.

    Greetings

    EB
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  6. #6
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    No i am looking for disks rotated by the x-axis.

    And also Im planning to draw the figure of the disks on the poster board. Im planning to cut 15 circles too stack on top of each other and decrease the radius all the way close to zero. My graph is 10 cm apart from 0 to 1. So can some one list the radiuses of the 15 circles.
    Last edited by Nimmy; June 6th 2006 at 07:06 PM.
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  7. #7
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    Quote Originally Posted by Nimmy
    No i am looking for disks rotated by the x-axis.

    And also Im planning to draw the figure of the disks on the poster board. Im planning to cut 15 circles too stack on top of each other and decrease the radius all the way close to zero. My graph is 10 cm apart from 0 to 1. So can some one list the radiuses of the 15 circles.
    Hello,

    If you want to draw 15 cylinders (a disk is a cylinder with a small height) it is advisible to make all cylinders with the same height.

    With your problem you get a height of \frac{1}{15}.

    You can calculate two different sets of cylinders: The smaller ones lay completely in a cone which is generated by the line y=-x+1,\ x\in [0;1], the greater ones lay completely outside of this cone. I've calculated both sets of radii. See attachment.

    (I assume, that you want to make a large drawing: So multiply the values of the radii by 10)

    The complete diagram looks like the 2nd attachment.

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails Riemann Sum Help-zyl_in_kegl.gif   Riemann Sum Help-zylinkegl_radii2.gif  
    Last edited by earboth; June 7th 2006 at 06:36 AM.
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