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Math Help - Sketch and Find the area of the region determined by the intersections of the curves.

  1. #1
    JDS
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    Sketch and Find the area of the region determined by the intersections of the curves.

    Sketch and Find the area of the region determined by the intersections of the curves.

    y= (2/(x2+1)), y=(absolute value of) x

    I do not know how to proceed. The book I am using is of no help. Perhaps a push in the right direction is in order?
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  2. #2
    JDS
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Well....I have given this one a go, and here is what I have come up with......

    My sketch.....

    Sketch and Find the area of the region determined by the intersections of the curves.-calculus-graph-sketch.png

    Here is my worked out problem....

    Sketch and Find the area of the region determined by the intersections of the curves.-my-calc-problem1.png

    If you can, let me know if I am right (or wrong)...Thanks!
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    That's not entirely correct. You need to split it into two separate integrals because the function on the right side is x and the function on the left side is -x.

    Technically the below is correct:

    \int_{-1}^{1} \frac{2}{1+x^2} - \left | x \right | dx

    But that's not what you did - you dropped the absolute value.
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    JDS
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Oh, I thought that I did seperate it into two different integrals...and I also thought the absolute value of x is simply x....
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    MHF Contributor MarkFL's Avatar
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    I would use the symmetry of the two even functions to state the area A of the region is:

    A=2\int_0^1\frac{2}{x^2+1}-x\,dx=\pi-1
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    JDS
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Quote Originally Posted by MarkFL2 View Post
    I would use the symmetry of the two even functions to state the area A of the region is:

    A=2\int_0^1\frac{2}{x^2+1}-x\,dx=\pi-1
    where did you come up with the \pi-1

    ......and also why are you evaluating the integral from 0 to 1......should it not be from -1 to 1 as I did in my example?

    Thanks in advance
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Quote Originally Posted by JDS View Post
    where did you come up with the \pi-1


    2\int_0^1\frac{2}{x^2+1}-x\,dx=\mathop {\left. {4\arctan (x) - x} \right|}\nolimits_{x = 0}^{x = 1}=?

    If g is an even function then \int_{ - a}^a {g(x)dx}  = 2\int_0^a {g(x)dx}
    Thanks from MarkFL
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Quote Originally Posted by JDS View Post
    where did you come up with the \pi-1

    ......and also why are you evaluating the integral from 0 to 1......should it not be from -1 to 1 as I did in my example?

    Thanks in advance
    You have already been told that in order to evaluate this area, you need to do TWO integrals, because the absolute value function is actually a HYBRID function, \displaystyle \begin{align*} |x| = \begin{cases} \phantom{-} x \textrm{ if } x \geq 0 \\ -x \textrm{ if } x < 0 \end{cases} \end{align*}, so you will need to perform an integral for EACH of those cases.
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    JDS
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Quote Originally Posted by Prove It View Post
    You have already been told that in order to evaluate this area, you need to do TWO integrals, because the absolute value function is actually a HYBRID function, \displaystyle \begin{align*} |x| = \begin{cases} \phantom{-} x \textrm{ if } x \geq 0 \\ -x \textrm{ if } x < 0 \end{cases} \end{align*}, so you will need to perform an integral for EACH of those cases.
    Thanks for your assistance...(and everyone thus far)... but I am not sure how to set it up to find those two different integrals, any advice?
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    MHF Contributor MarkFL's Avatar
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Hint: |x| = -x when x < 0 and |x| = x when 0 ≤ x.
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  11. #11
    JDS
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    Re: Sketch and Find the area of the region determined by the intersections of the cur

    Thanks everyone, here is my solution.

    Sketch and Find the area of the region determined by the intersections of the curves.-problem10.png

    NOTE: I wanted to make sure and give much thanks to Jacek! Jacek has been tutoring me in his spare time and is an Excellent Teacher!!! Thank you Jacek!
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