I'm teaching the section 4.7 on optimization in Stewart Calculus. It has a subsection on "Applications to Business and Economics." There the author defines the price function $p(x)$ to be the price charged if $x$ items have sold, and the revenue function $R(x)$ by the formula $R(x)=xp(x)$.
This seems like an absurd definition to me. For if the demand drops very sharply, say $p(x)=e^{-x}$, then for large $x$, $R(x)$ is approximately 0. In this case the names would suggest the company made nothing, even though they sold many items.
Do people actually use this definition? It seems to me the correct definition would be $R(x)=\int_{0}^{x}p(s)ds$. However, checking the first few results on Google, I only found the given definition. Is my thinking way off-base here?
This is an economics question rather than mathematics.
There is a principle of "the law of one price" which says that in efficient markets each item sells for the same price as other identical items.
When this applies, the revenue is clearly the number of items sold multiplied by the price per unit.
In your example, it makes no sense for a seller to offer more items for sale than there is demand and then to reduce the price so that all the items are sold. With a price of $p(x)=e^{-x}$, the revenue $R(x)=xp(x)$ is maximised when $x=1$ and the seller has no incentive to offer more than this for sale: the optimal amount may be less if the seller also has costs.