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Math Help - Evaluating integrals

  1. #1
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    Evaluating integrals

    I've attached a picture of the problem I'm working on involving evaluating integrals, I know how to do it visually (if I have a drawn graph in front of me or can draw a graph) which is how I got the correct answer to #1 that you see there, but when the numbers get too huge too simply graph I have no idea how to go about solving these kinds of problems, I've looked through my notes and book and can't make sense of all the steps they're giving me, could someone help me out?
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    I apologize for the double post but I figured it'd be better to post this related tidbit here rather than start a whole new thread

    What about evaluating integrals from pictures of graphs like this other picture I've attached below, I thought I had it down because the area of a circle is just pi r^2 and I did that and took half of it since it's a half circle, why doesn't that work here, and what should I do?
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  3. #3
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    Quote Originally Posted by fattydq View Post
    I've attached a picture of the problem I'm working on involving evaluating integrals, I know how to do it visually (if I have a drawn graph in front of me or can draw a graph) which is how I got the correct answer to #1 that you see there, but when the numbers get too huge too simply graph I have no idea how to go about solving these kinds of problems, I've looked through my notes and book and can't make sense of all the steps they're giving me, could someone help me out?
    Hi

    For the second exercise \Delta x = \frac{6}{n} and x_i = 1 + \frac{6}{n}\:i

    \sum_{i=1}^{n}f(x_i) \Delta x = \frac{6}{n} \sum_{i=1}^{n}\left( \left(1 + \frac{6}{n}\:i\right)^2 + 2 \left(1 + \frac{6}{n}\:i\right) - 7 \right)

    \sum_{i=1}^{n}f(x_i) \Delta x = 120 + \frac{180}{n} + \frac{36}{n^2}

    The integral is the limit when n tends towards +\infty therefore 120
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  4. #4
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    Quote Originally Posted by fattydq View Post
    I've attached a picture of the problem I'm working on involving evaluating integrals, I know how to do it visually (if I have a drawn graph in front of me or can draw a graph) which is how I got the correct answer to #1 that you see there, but when the numbers get too huge too simply graph I have no idea how to go about solving these kinds of problems, I've looked through my notes and book and can't make sense of all the steps they're giving me, could someone help me out?
    For your second problem

    \Delta x = \frac{7-1}{n} = \frac{6}{n} and x_i = 1 + \frac{6}{n} i

    so the sum is

    \lim_{n \to \infty} \sum_{i=1}^n \left[ \left( 1 + \frac{6}{n} i \right)^2 + 2 \left(1 + \frac{6}{n} i \right) - 7 \right] \frac{6}{n}

    after expanding

    6 \lim_{n \to \infty} \sum_{i=1}^n \frac{36 i^2}{n^3} + \frac{24i}{n^2} - \frac{4}{n}

    distributing the sum

    6 \lim_{n \to \infty} \left( \frac{36}{n^3} \sum_{i=1}^n i^2 + \frac{24}{n^2} \sum_{i=1}^n i - \frac{4}{n} \sum_{i=1}^n 1 \right)

    substituting in the formulas for the sums, expanding and taking the limit gives

    6 \lim_{n \to \infty} \left( \frac{36}{n^3} \frac{n(n+1)(2n+1)}{6} + \frac{24}{n^2} \frac{n(n+1)}{2} - \frac{4}{n} n \right) = 120
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  5. #5
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    Quote Originally Posted by running-gag View Post
    Hi

    For the second exercise \Delta x = \frac{6}{n} and x_i = 1 + \frac{6}{n}\:i

    \sum_{i=1}^{n}f(x_i) \Delta x = \frac{6}{n} \sum_{i=1}^{n}\left( \left(1 + \frac{6}{n}\:i\right)^2 + 2 \left(1 + \frac{6}{n}\:i\right) - 7 \right)

    \sum_{i=1}^{n}f(x_i) \Delta x = 120 + \frac{180}{n} + \frac{36}{n^2}

    The integral is the limit when n tends towards +\infty therefore 120
    I don't understand at all how either of you got from setting the problem up to the next step where it suddenly equals 120+whatever+whatever. In other words, including "Hi" as a line, I don't understand how you got from line 3 of your work to line 4
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  6. #6
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    He just set up the sum and after some algebraic expansion he applied properties of sums and that's all. You should know that.
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  7. #7
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    Quote Originally Posted by Krizalid View Post
    He just set up the sum and after some algebraic expansion he applied properties of sums and that's all. You should know that.
    I must not, because it makes no sense how he went from (1+6i/n)^2 to 120, etc etc
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  8. #8
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    Wait, does it have something to do with the n(n+1)/2 formula for summation? It seems like it's connected somehow I just don't know how
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  9. #9
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    Quote Originally Posted by danny arrigo View Post
    For your second problem

    \Delta x = \frac{7-1}{n} = \frac{6}{n} and x_i = 1 + \frac{6}{n} i

    so the sum is

    \lim_{n \to \infty} \sum_{i=1}^n \left[ \left( 1 + \frac{6}{n} i \right)^2 + 2 \left(1 + \frac{6}{n} i \right) - 7 \right] \frac{6}{n} (1)

    after expanding

    6 \lim_{n \to \infty} \sum_{i=1}^n \frac{36 i^2}{n^3} + \frac{24i}{n^2} - \frac{4}{n} (2)

    distributing the sum

    6 \lim_{n \to \infty} \left( \frac{36}{n^3} \sum_{i=1}^n i^2 + \frac{24}{n^2} \sum_{i=1}^n i - \frac{4}{n} \sum_{i=1}^n 1 \right) (3)

    substituting in the formulas for the sums, expanding and taking the limit gives

    6 \lim_{n \to \infty} \left( \frac{36}{n^3} \frac{n(n+1)(2n+1)}{6} + \frac{24}{n^2} \frac{n(n+1)}{2} - \frac{4}{n} n \right) = 120 (4)
    In the four step in red, which one confuses you?
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  10. #10
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    Quote Originally Posted by danny arrigo View Post
    In the four step in red, which one confuses you?
    Well in step 2 for starters you pull the 6 out in front of everything but not the whole 6/n, I don't understand how/why that is. Also in step 2 I don't know how you got the 24i over n^2 and -4/n, I know the 36i^2 comes from 6i being squared but don't understand how you got to the other two terms.

    And since there's all that confusion that early on, the rest of your steps subsequently make no sense to me.
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