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Anti-Diff and Differential Equations help.

  1. May 13, 2007 #1
    --------------------------------------------------------------------------------

    Hey, theres a few questions i need help with urgently, would apreciate any help on as many of them as fast as possible. Thanks in advance.

    1.
    Find the area enclosed by the curves:
    y² = 4 + x and x + 2y = 4


    2.
    Fine the volume of the solid of revolution generated by rotating about the y-axis the region enclosed by the y-axis and the curves y = 2x^2 - 1 and y =
    √(x)


    3.
    Use the method of cylindrical shells to find the volume of the solid of revolution generated by rotating about the y-axis the region enclosed by y =
    √(x+1), y = 0, x=0 and x=4


    Differential equations now:

    4.
    Solve
    dy/dx = (x(y²+3)/y

    5.
    Solve
    dy/dx = e^(x-2y), y(1) = 0

    6.
    dy/dx = x²y-2e^(x), y(1) = -1

    Use Euler method with 4 stephs to find and approx value for y(2)




    Thank you sooo much for the help
     
  2. jcsd
  3. May 13, 2007 #2

    Hootenanny

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    Gold Member

    Welcome to the forums,

    According to our guidelines, one is expected to show one's own efforts before asking for help.
     
  4. May 13, 2007 #3

    cristo

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    Welcome to PF Jozsa. Please note that you are rquired to show some work before we can help you. So, what have you attempted for the questions thus far?

    [Damn.. beaten to it again! I'll just go back to my own revision then! :wink: ]
     
  5. May 13, 2007 #4

    Hootenanny

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    Your making me feel guilty now... I can revise and post at the same time... right?:rolleyes:

    [best of luck with your exams cristo!]
     
  6. May 13, 2007 #5

    cristo

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    Sorry... yea sure you can! :biggrin:
    Thanks; good luck in your remaining exams!
     
  7. May 13, 2007 #6
    Okay Update,



    1.

    Find the area enclosed by the curves:

    y² = 4 + x and x + 2y = 4



    I worked it out myself, so dont worry









    2.

    Fine the volume of the solid of revolution generated by rotating about the y-axis the region enclosed by the y-axis and the curves y = 2x^2 - 1 and y =

    √(x)



    V = integral 1 to 0 of ( pi (√(x) - 2x^2 - 1)^2 )



    So i expanded (√(x) - 2x^2 - 1)^2



    to get (simplified): 4x^4 - 4x^2.5 - 4x^2 + 2√(x) + x + 1



    then got the integral is:



    pi((4/5)x^5 - (4/3.5)x^3.5 - (4/3)x^3 + (2/1.5)x^1.5 + (x^2)/2 + x) between 0.42 and 0



    so subbing in those values: pi((81/70) - 0)



    = 3.64 units^3



    Correct?







    3.

    Use the method of cylindrical shells to find the volume of the solid of revolution generated by rotating about the y-axis the region enclosed by y = √(x+1), y = 0, x=0 and x=4



    V = integral_{4 to 0} of [ 2pi x ( √(x)+1 ) ] dx



    = (2pi)integral_{4 to 0} of (x^1.5 + x) dx



    = (2pi)[(x^2.5)/2.5 + (x^2)/2] between 4 and 0



    = (2pi)(12.8 + 8)



    = 41.6 pi units^3



    Correct?





    4.

    Solve

    dy/dx = (x(y²+3)/y)



    (y/(y^2+3)) dy/dx = x



    so: integral (y/(y^2+3)) dy = integral x dx + C



    so the integral is: (1/2)ln(y^2+3) = x^2/2 + C



    then, we say ln(y^2+3) = x^2 + A

    where A = 2C



    so, y^2 = = e^(x^2 + A) - 3



    y = √(e^(x^2 + A) - 3)



    Correct?







    5.

    Solve

    dy/dx = e^(x-2y), y(1) = 0



    ok.. so im pretty much stumped after this step



    ln(dy) - ln(dx) = x - 2y



    ln(dy) + 2y = ln(dx) + x



    Help?







    6.

    dy/dx = x²y-2e^(x), y(1) = -1

    Use Euler method with 4 steps to find and approx value for y(2)



    The formula for calculating succcessive values of x and y is :

    y_n+1 = y_n + 0.25[(x_n^2)(y_n)-2e^(x_n)]



    (_n+1 means subscript n+1 etc)



    y_1 = -1 + 0.25[(1^2)(-1)-2e^(1)] = -2.6091

    y_2 = -2.6091 + 0.25[(1.25^2)(-2.6091)-2e^(1.25)] = -5.3735

    y_3 = -5.3735 + 0.25[(1.50^2)(-5.3735)-2e^(1.50)] = -10.6369

    y_4 = -10.6369 + 0.25[(1.75^2)(-10.6369)-2e^(1.75)] = -21.6581





    So the value at y(2) = -21.6581



    Yes? Correct?
     
    Last edited: May 13, 2007
  8. May 13, 2007 #7
    Anyone? Help!!!
     
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