How Many Fragments Are Found Within 10 Kilometers of a Volcanic Eruption?

[radeksrat]

Not really a homework question, but a problem I don't get nonetheless.



The density of fragments lying x kilometers from the center of a volcanic eruption is given by:

D(r) = 1/[sqrt(x) +2] fragments per square kilometer. To 3 decimal places, how many fragments will be found within 10 kilometers of the eruption's center?



I thought I was supposed to integrate the function from 0 to 100*pi, and in doing so I got 26.294, (I got 2[sqrt(x) - 2*ln(sqrt(x)+2)] when i integrated the function) but the answer was given to me as 70.424. The answer could very well be wrong, but I don't know that it is. What, if anything, am I doing wrong here?

 

[abercrombiems02]

in cylindrical coordinates the integral of the density gives the distribution.
In this case the problem requires integrating over an area thus we have a double integral. In polar form J = int(int(f(r,theta)*r*dr)*dtheta) With the appropriate limits. Then J = int(int(1/(sqrt(x)+2),x) from 0 to 10,theta from 0 to 2pi) The result is simpler because theta does not appear inside the integral. The result is 2*pi*int(1/(sqrt(x)+2),x) from 0 to 10. That should be your answer

 

[0rthodontist]

I'm not sure what you mean by ,x) abercrombie, but to be clear, the integral is
$$\int_{0}^{2 \pi}\int_{0}^{10} \frac{x}{\sqrt{x}+2} dx d\theta$$
because of the jacobian x.