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For the following scenario,

The density of people in a 200-ft long stadium during a concert is given by c(x), where x is the distance, in feet, from the stage. Find the number of people at the concert who are at most x feet away from the stage and express the answer as a Riemann sum.

I came up with the definite integral for the number of people as $\int_{0}^{x} c(x)$

I am wondering if $\int_{0}^{x} c(x)$ is equivalent to the following Riemann sum for finding the number of people at the concert:

$$\sum_{k=1}^{n} f(x_k) \space \Delta x$$

Thanks for your time.

Tom
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  • What is $f$? If by $f$, you mean $c$, then the integral is the limit of the Riemann sums as $n\to \infty$. – J126 May 11 '13 at 15:32
  • Stadium seems to implies that the people are distributed radially around the stage. In that case, the integral should be $\int 2\pi xc(x)$, since you're effectively counting people in concentric rings around the stage, and the ring with radius $r$, width $w << r$ and density $c(r)$ contains approximately $2\pi w r c(r)$ people. – fgp May 11 '13 at 15:37
  • bettor to use different variable for the variable of integration and the variable of upper/lower limit of integral as in $\int_0^Dc(x) dx$. For your Riemann sum somehow you need to make it clear that the $n$th interval ends at distance $D$ from stage. Better yet think of an explicit instruction to be given to someone to do the counting and convert that to a set of formulas. – Maesumi May 11 '13 at 15:39
  • Thanks for the suggestions and help everyone. The original question wasn't very specific on the actual shape of the stadium, and just provided some possible equations for the number of people. It did not specify if $f$ was $c$ either. – Tom May 11 '13 at 15:51

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