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Thread: Estimate The Value Of Integral

  1. #1

    Lightbulb Estimate The Value Of Integral

    Hi Everyone,

    I am having trouble getting started on this problem.

    Book Problem 47

    Given that [tex]\,3\, \leq\, f(x)\, \leq\, 6\,[/tex] for [tex]\, -8\, \leq\, x\, \leq\, 8,\, [/tex] estimate the value of [tex]\, \int_{-8}^{8}\, f(x)\, dx[/tex]

    I honestly have no idea where to go with this one. Any help would be appreciated.
    Last edited by stapel; 07-06-2015 at 08:10 PM. Reason: Typing out the text contained in the graphic.

  2. #2
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    Well, you know that the integral can be represented by the area between the graph of the function and the x-axis. So, then, to find the upper bound of the integral, you have to find a function that satisfies the domain and range with the most area underneath it. How would you go about that? And similarly, for the lower bound, can you find the function with the least area underneath it? I recommend sketching several possible functions which have the given domain and range to see if you can find a pattern.

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    Quote Originally Posted by ksdhart View Post
    Well, you know that the integral can be represented by the area between the graph of the function and the x-axis.
    That is not always true.
    Consider [tex]\displaystyle\int_0^\pi {\cos (t)dt} = 0[/tex]. But the bounded area is two.
    Last edited by pka; 07-07-2015 at 06:38 PM.
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  4. #4
    Elite Member stapel's Avatar
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    Quote Originally Posted by chrisalau32 View Post
    Book Problem 47

    Given that [tex]\,3\, \leq\, f(x)\, \leq\, 6\,[/tex] for [tex]\, -8\, \leq\, x\, \leq\, 8,\, [/tex] estimate the value of [tex]\, \int_{-8}^{8}\, f(x)\, dx[/tex]

    I honestly have no idea where to go with this one.
    What is the lowest value that f(x) can be? If f(x) is this lowest value over the whole length (from x = -8 to x = 8), what would be the area?

    What is the highest value that f(x) can be? If f(x) is this highest value over the whole length, what would be the area?

    So what are the bounds on the area?

    If you get stuck, please reply with your answers to the questions above. Thank you!

  5. #5
    Quote Originally Posted by stapel View Post
    What is the lowest value that f(x) can be? If f(x) is this lowest value over the whole length (from x = -8 to x = 8), what would be the area?

    What is the highest value that f(x) can be? If f(x) is this highest value over the whole length, what would be the area?

    So what are the bounds on the area?

    If you get stuck, please reply with your answers to the questions above. Thank you!

    The lowest value f(x) could be is 3 and the highest that it could be is 6 but I don't see how that helps me with this problem. Not saying you are wrong, just saying I don't really understand.

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    Quote Originally Posted by chrisalau32 View Post
    The lowest value f(x) could be is 3 and the highest that it could be is 6 but I don't see how that helps me with this problem. Not saying you are wrong, just saying I don't really understand.
    I don't know why this question irritates me so, but is does. You should have been given this theorem.

    If each of [tex]g~\&~h[/tex] is an intergable function on [tex][a,b][/tex] and if [tex]\left( {\forall x \in \left[ {a,b} \right]} \right)\left[ {g(x) \le h(x)} \right][/tex] then [tex]\displaystyle\int_a^b {g(x)dx} \le \int_a^b {h(x)dx} [/tex].

    Thus your answer is [tex]\displaystyle 3\cdot 16\le\int_{-8}^8 {f(x)dx} \le 6\cdot 16 [/tex]
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

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    I find that to solve problems - I need to sketch the information.

    If you do that - the theorem that pka is talking about (and the solution) would basically jump out.

    Knowing that the maximum value of f(x) = 6 and the minimum value is 3 then

    [tex]\displaystyle{\int_{-8}^{8}f(x)|_{min}dx \ \le \ \int_{-8}^{8}f(x)dx \le \ \int_{-8}^{8}f(x)|_{max}dx }[/tex]

    = [tex]\ \displaystyle{\int_{-8}^{8}3 \ dx \ \le \ \int_{-8}^{8}f(x)dx \le \ \int_{-8}^{8}6 \ dx }[/tex]
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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    Elite Member stapel's Avatar
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    Quote Originally Posted by chrisalau32 View Post
    The lowest value f(x) could be is 3 and the highest that it could be is 6 but I don't see how that helps me with this problem. Not saying you are wrong, just saying I don't really understand.
    Okay. If the function is always at the lowest value, then what shape is the area? How wide is the area? What is the height of the area? What then is the area?

    If the function is always at the highest value, then what shape is the area? How wide is the area? What is the height of the area? When then is the area?

    Since the function is always somewhere between the low and high values (including "or equal to" those values), what then are the bounds on the area?

    Please reply with your thoughts and answers. Thank you!

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