Find the total revenue , calculus The Marginal Revenue of an Item sold is $ \ R(x)= 100 e^{-0.001 x} \ $ dollers . 
Find the total revenue by selling items $ \ 101 \ $ through $ \ 1000 \ $. 
Answer:
I think this would be the integral  $ \ \int_{101}^{1000} R(x) \ dx \ \ $
But I am not sure .
Help me 
 A: As the comments stated, this is a basic algebra problem. Nonetheless, it requires some attention to details.
As an arithmetic series, denote $\displaystyle S \equiv \sum_{x = 101}^{1000}R(x)~$, then
\begin{align*}
S = 100 \sum_{x = 101}^{1000} e^{\frac{-x}{1000}} &= 100 \,e^{\frac{-100}{1000}}\sum_{x = 1}^{900} e^{\frac{-x}{1000}} \\
&= 100 \,e^{\frac{-1}{10}} \frac{ e^{\frac{-1}{1000} } - e^{\frac{-901}{1000} } }{1 -  e^{\frac{-1}{1000}} }  \qquad \text{as}~ \sum_{x = 1}^{900} r^x = \frac{ r - r^{901} }{1 - r} \quad \text{where}~ r = e^{\frac{-1}{1000}}
\end{align*}
The numerical value is about $53669$, or with more digits, $53668.95426225823\ldots$ as per Wolfram Alpha or any computing software.
Let's compare this with the following integral approximations, all courtesy of Wolfram Alpha:
Original poster's suggestion, integration limits nominally match: $\displaystyle\int_{x = 101}^{1000}R(x) \approx 53605$
Commented by @cgiovanardi: standard first order correction 
 $\displaystyle\int_{x = 100.5}^{1000.5}R(x) \approx 53669$ , very close to the exact solution.
Just shift it for the sake of comparison to the right: $\displaystyle\int_{x = 101}^{1001}R(x) \approx 53642$
Shift for the sake of comparison to the left: $\displaystyle\int_{x = 100}^{1000}R(x) \approx 53696$
