Integral of a function + limit

In summary: So the answer is "N → ∞".plotting a graph of this Equation over a longer time scale (millions of years) would be a good idea.
  • #1
Bikkehaug
6
0

Homework Statement



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.

Homework Equations




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?
 
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  • #2
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.
 
  • #3
Bikkehaug said:

Homework Statement



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.

Homework Equations




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.

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

Does anyone have any idea about how I can proceed from this?

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

Related to Integral of a function + limit

What is an integral of a function?

The integral of a function is a mathematical concept that represents the area under the curve of a function on a given interval. It is denoted by the symbol ∫ and is used to find the total accumulation of a quantity over a range of values.

What is a limit of a function?

The limit of a function is the value that a function approaches as the input variable gets closer and closer to a specific value. It is used to describe the behavior of a function at a particular point and is denoted by the symbol lim.

What is the relationship between an integral and a limit?

The relationship between an integral and a limit is that the integral of a function can be evaluated using limits. Specifically, the integral of a function is equal to the limit of a sum of infinitely many rectangles as the width of the rectangles approaches zero.

How do you calculate the integral of a function using limits?

To calculate the integral of a function using limits, you can use the fundamental theorem of calculus, which states that the integral of a function can be evaluated by finding its antiderivative and evaluating it at the upper and lower limits of integration.

What is the significance of the integral of a function + limit in science?

The integral of a function + limit is significant in science because it allows us to calculate important quantities such as distance, velocity, and acceleration. It is also used in many physical and engineering applications, such as calculating the work done by a force or the area under a velocity-time graph.

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