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Integral of a function + limit

  1. Mar 27, 2014 #1
    1. The problem statement, all variables and given/known data

    Plutonium is a radioactive waste.. etc. A mass og 1.000 kg will after x years be reduced to:

    m(x) = 1.000 * e-2.89*10-5x

    The yearly Waste of plutonium is 1kg. The total plutonium mass after N years is given by:

    ∫m(x)dx where the upper value of the integral is N, and the lower value is 0 (not sure how to Write this directly into the formula)

    Compute the integral, and estimate the total waste of plutonium after a long time.

    2. Relevant equations


    3. The attempt at a solution

    Solving the integral: ∫1.000 * e-2.89*10-5x dx

    = [[itex]\frac{1}{-2.89*10^-5}[/itex] * e-2.89*10-5x ] Again, the higher value is N, the lower value is 0.

    If I plot a graph of this Equation, I get a linear graph With the mass after, say 5000 years is 5000kg. After 3 years the mass is 3kg.

    Does anyone have any idea about how I can proceed from this?
     
  2. jcsd
  3. Mar 27, 2014 #2

    jbunniii

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    Try plotting over a longer time scale (millions of years). Unfortunately, ##5000## years is negligible due to the ##-2.89\times 10^{-5}## coefficient. One way to understand why you are seeing a straight line is to write ##\alpha = -2.89\times 10^{-5}##. Then for values of ##x## such that ##\alpha x## is small, we can approximate ##e^{ax} \approx 1 + ax##, so the result of your integration can be approximated as
    $$\left.\frac{1}{\alpha}(1 + ax)\right|_{0}^{N} = \left.\left(\frac{1}{\alpha} + x\right)\right|_{0}^{N} = N$$
    So the growth is approximately linear until ##x## is large enough that this approximation no longer holds.
     
  4. Mar 27, 2014 #3

    pasmith

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    So that's
    [tex]
    \frac{1}{2.89 \times 10^{-5}} \left(1 - e^{-2.89 \times 10^{-5} N}\right).
    [/tex]

    "estimate ... after a long time" means "let [itex]N \to \infty[/itex]".
     
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