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March 29, 2017

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does anyone know how to approximate the are and circumfrence of the region bounded by the given curves
y=cos(x^2 +(100493/100000)), y=1+x-X^2

  • calculus wxmaxima - ,

    The two curves intersect at
    -0.4425 and 1.8018.

    By integrating (numerically)
    f(x)=cos(x^2 +(100493/100000)) - (1+x-x^2)
    between the two roots, you would get the area as 1.99326.

    The perimeter can be obtained by integrating the curve length of the separate curves between the same limits.

    See for arc-length:
    http://en.wikipedia.org/wiki/Arc_length

  • calculus graph - ,

    For graph of function, see:
    http://img20.imageshack.us/i/1297900781.png/

  • calculus wxmaxima - ,

    Thank you very much but i am still confused with the arc length or the circumference

  • calculus wxmaxima - ,

    To find the curve length, we only need to integrate ds (incremental distance) between the given limits.

    The expression of ds is basically the slant distance obtained by sqrt(dx²+dy²), so the integral is:
    ∫sqrt(1+(dy/dx)²dx
    It gets a little messy unless the function is a polynomial, but calculating using numerical integration is rather easy, even for complicated functions.

    For the cosine part of the function, the derivative should be done using the chain rule, and you should get:
    dy/dx = -sin(x^2 +(100493/100000)) * 2x
    Apply the integral above and integrate from -0.44 to 1.8 (approx.) to get about 3.4 as the arc length.

    Check: Δx = 1.8-(-0.44)=2.24
    3.4/2.24=1.5 approx. which is about sec(45°).

    If you have other questions, just post.

  • calculus wxmaxima - ,

    i understand but i just don't know how to plug it in wxmaxima is it possible if you can show me
    thanks very much

  • calculus wxmaxima - ,

    Define a function f(x) (to find arc-length)
    f(x):=......
    Find its derivative:
    diff(f(x),x)
    Copy and paste the result and assign it to another function, say f1(x):
    f1(x):=....paste....
    Go to the Calculus/integrate function.
    For the integrand, input:
    sqrt(1+f1(x)^2)
    Input the limits of integration.
    check the 'numerical integration' box.
    leave the method as default, Romberg or the other one will both work. Best is try both and compare. This will give you an idea of the accuracy.
    Click OK and wait.

    Post if you could use further help.

  • calculus wxmaxima - ,

    okay so for the first f(x) i plug in cos(x^2 +(100493/100000))find the derivative then do i do the same for y=y=1+x-X^2 ?

  • calculus wxmaxima - ,

    Exactly! Repeat all the steps for the polynomial.

    Since the two functions are to be integrated within the same interval, you are better off doing them separately.

  • calculus wxmaxima - ,

    okay so i put y=1+x-X^2 found the derivative 1-2*x then i pluged that in into sqrt(1+1-2*x)^2) then i did the romberg thing so is this right i got -1.4336 for this one

  • calculus wxmaxima - ,

    WEll THANKSSSS ALOTTTTTTTTTTTTTTTTT!

  • calculus wxmaxima - ,

    Did your length for the cos function match mine at about 3.4?

    Glad that everything worked out!

    À la prochaine!

  • calculus wxmaxima - ,

    Oops, I don't think -1.4 is right for the polynomial.
    Did you use sqrt(1+(dy/dx)^2)?
    Your expression sqrt(1+1-2*x)^2) has a missing parenthesis.
    I got 2.29 for the polynomial, and 3.4 for the cos function.

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