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Math Help - How do I do these Calculus Problems?

  1. #1
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    How do I do these Calculus Problems?

    I need to make an integral that works for all of this...
    (1^2+2^2+3^2+4^2+...+n^2)/(n^3)

    I need to find the limit as x approaches 5 of this: (2x/(x-5)) x the integral of [sin(t)]/t dt.
    I think this has something to do with as f(a)-f(x)/(a-x)=f'(a) or something like that.


    Last one I need help on is the integral of square root of (b^2-x^2) db from the interval (b/square root of 2) to b. I need to show the graph without using a calculator or at least show where it should be.

    You don't have to answer all questions but 1 would be nice.
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  2. #2
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    Quote Originally Posted by DaAzNJRiCh View Post
    I need to make an integral that works for all of this...
    (1^2+2^2+3^2+4^2+...+n^2)/(n^3)

    I need to find the limit as x approaches 5 of this: (2x/(x-5)) x the integral of [sin(t)]/t dt.
    I think this has something to do with as f(a)-f(x)/(a-x)=f'(a) or something like that.


    Last one I need help on is the integral of square root of (b^2-x^2) db from the interval (b/square root of 2) to b. I need to show the graph without using a calculator or at least show where it should be.

    You don't have to answer all questions but 1 would be nice.
    This is confusing.

    What are you asking for in the first question? It looks to me like you want a closed form expression for the summation? ( (n+1)(2n+1)/(6n^2) )

    The integral has nothing to do with x, so the limit does not exist. I'm having a hard time understanding why this question was asked because the answer is too simple.

    I can give you a hint as to how to do the integral (sub b = x*sec(theta)), but what do you mean by a graph? The integral is a real number.

    -Dan
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  3. #3
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    Hello, DaAzNJRiCh!

    Like Dan, I don't understand these problems . . . at all.


    I need to make an integral that works for all of this...
    . . (1 + 2 + 3 + ... + n)/n

    Could you give us the original wording of the problem?

    We can simplify that expression:
    . . n(n + 1)(2n + 1)/6n .= .(n + 1)(2n + 1)/6n

    but I don't see how we can "make" an integral.



    The integral of square root of (b^2-x^2) db from the interval (b/square root of 2) to b.
    I need to show the graph without using a calculator or at least show where it should be.

    First of all, the integral should have dx . . . right?
    . . It appears that b is a constant.
    . . . . . . . . . . . . . . .______
    The graph of: .y .= .√b - x .is a semicircle with center (0,0) and radius b.
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._
    And we want the area under the curve from b/√2 to b.

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  4. #4
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    To make an integral use f(x)=x^2 on [0,1].
    Using right-hand Riemann Sums with delta(x)=1/n .
    Last edited by Plato; October 10th 2006 at 07:10 PM.
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  5. #5
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    1.I need it so that is an integral like the integral of f(x) from 1 point to another. Something like that.

    2. I need to find the limit as x approaches 5 of this: (2x/(x-5)) times the integral of [sin(t)]/t dt.


    3. And it is dx I think but I need to find the value as well.
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  6. #6
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    Quote Originally Posted by DaAzNJRiCh View Post
    2. I need to find the limit as x approaches 5 of this: (2x/(x-5)) times the integral of [sin(t)]/t dt.
    By sin(t)/t you mean,
    g(x)=INTEGRAL^x_1 sin(t)/t dt
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