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Thread: Calculating area from a graph.

  1. #1
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    Calculating area from a graph.

    I have to calculate the area of the shaded graph (in the link). Seeming as no equations are given, how do I calculate it?

    Untitled | Flickr - Photo Sharing!

    It's hard to read the graph but it's just 0.1π, 0.2π... 2.0π.
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  2. #2
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    To me it seems like it is a sine and cosine graph.

    You would need to develop an equation to be able to calculate the area accurately:

    $\displaystyle y = a*sin(bx)$
    $\displaystyle y = a * cos(bx)$

    Where a and b are constant values.
    To me it seems its maximum is 4.5 and minimum is -4.5. Thus constant a must be 4.5.
    One full cycle seems to be 2.0π. If you are working in radians, constant b must be $\displaystyle 2\pi$/2. If working in degrees, b = 360/2.


    $\displaystyle y = 4.5*sin(\pi {x})$
    $\displaystyle y = 4.5 * cos(\pi {x})$
    (For working in radians)

    There once you have your 2 equations, you can integrate between 0 and 2 to find the area.
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  3. #3
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    Of course! Thankyou so much.
    Can I just ask, how do you know to integrate between 0 and 2?
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  4. #4
    Senior Member Educated's Avatar
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    Oh sorry I misinterpereted the figures on your graph.

    Now I take a closer look, 2.0π actually meant $\displaystyle 2\pi$. I thought π was the units...

    Forget what I said about b being a constant. It stretches out like a normal graph - from 0 to $\displaystyle 2\pi$.

    Thus:
    $\displaystyle y = 4.5*sin(x)$
    $\displaystyle y = 4.5 * cos (x)$

    To make finding the area of the shaded part easier, move the graph up by 4.5 so that no part of it is in the negative part, below the x-axis.


    $\displaystyle y = 4.5*sin(x)+4.5$
    $\displaystyle y = 4.5 * cos (x)+4.5$

    To find the area:
    Integrate between 0 to $\displaystyle \frac{1}{4}\pi$, and do the cosine graph subtracted the sine graph.
    Then integrate between $\displaystyle \frac{1}{4}\pi$ to $\displaystyle \frac{5}{4}\pi$, and do the sine graph subtracted the cosine graph this time.
    Finally integrate between $\displaystyle \frac{5}{4}\pi$ to $\displaystyle \frac{8}{4}\pi$, and do the cosine graph subtracted the sine graph.
    Add all of these together and you will find your shaded area.
    Last edited by Educated; Aug 21st 2010 at 10:28 PM. Reason: Didn't see the complexity of the problem, I wasn't thinking before.
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  5. #5
    Senior Member Educated's Avatar
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    Also the sine and cosine graphs go onto infinity - I'm only guessing the shaded parts are only between 0 to $\displaystyle 2\pi$ as that is as far as the graph goes.
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