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Math Help - URGENT MATH HELP: Differntial Equations and Applications of Intergration

  1. #1
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    URGENT MATH HELP: Differntial Equations and Applications of Intergration

    2. A culture starts with 200 bacteria and grows at a rate proportional to its size. After 3 hours there are 3600 bacteria. Find the rate of growth after 4 hours.

    3. Find the Area of the region bound by f(x)=2-x^2 and g(x)=x^2 by first simplifying the integral based upon whether the integrand is an even or an odd function. I know how to find even and odd functions but how do you simplify in terms of integrals?

    4. Set up an integral that gives the area of the region bounded by x=y^2-3 and y=x/2.
    My answer is evaluating the integral from -3 to 6 for the function x=y^2-3 and subtracting the integral for the second function (y=x/2) evaluated from 0 to 6. Am I on the right track?

    Any help at all would be much appreciated. Thank you ahead of time.
    Last edited by PMoNEY23; May 26th 2009 at 12:02 AM.
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  2. #2
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    Quote Originally Posted by PMoNEY23 View Post
    1.Find the equation of the curve that passes through point (1,3) and has a slope of y/x^2 at any point (x,y).
    \frac{dy}{dx}=\frac{y}{x^2} so \frac{dy}{y}=\frac{dx}{x^2}

    Integrate both sides yeilds ln|y|=-\frac{1}{x}+c now we can use the condition to find what c is

    ln|3|=-\frac{1}{1}+c so c=ln3+1

    And now exponentiate with base e to get

    |y|=e^{-\frac{1}{x}}e^c=3e^{1-\frac{1}{x}}

    Notice that if we plug in x=1, e get y=+3, so we can remove the absolute value condition
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