Why does Revenue = Units Sold * Demand? Wouldn't units sold depend on demand?

In https://youtu.be/VfmLrGSdfJI?t=22m21s why does revenue = units sold * demand? I don't understand how this is determined. Wouldn't units sold depend on demand?
This video seems to me to be truly terrible. It is way too fast; I can't even carefully study the problem before it is off the screen. And it is quite sloppy: using P to stand for profit and p to stand for price is an invitation to confusion, particularly in a lecture where you must process both by ear and eye simultaneously. And I think that, once or twice, he idiotically uses "demand" when he intends "revenue." (Does he really say that profit = demand less cost or did I dream that?) I think he has also confused "sales," usually considered a monetary figure, with "units sold," an unambiguously non-monetary figure. Perhaps somewhere in this mess he actually defines his variables in a way that makes economic and accounting sense, but I missed it if he did (or he did it before where you started the video.)

But his expression for revenue of xp simply means quantity sold as a function of time times price per unit sold. And the problem is about what is the rate of change in units sold given the rate of change in profits under the assumption that price per unit and variable cost per unit do not change over time.
 
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This video seems to me to be truly terrible. It is way too fast; I can't even carefully study the problem before it is off the screen. And it is quite sloppy: using P to stand for profit and p to stand for price is an invitation to confusion, particularly in a lecture where you must process both by ear and eye simultaneously. And I think that, once or twice, he idiotically uses "demand" when he intends "revenue." (Does he really say that profit = demand less cost or did I dream that?) I think he has also confused "sales," usually considered a monetary figure, with "units sold," an unambiguously non-monetary figure. Perhaps somewhere in this mess he actually defines his variables in a way that makes economic and accounting sense, but I missed it if he did (or he did it before where you started the video.)

But his expression for revenue of xp simply means quantity sold as a function of time times price per unit sold. And the problem is about what is the rate of change in units sold given the rate of change in profits under the assumption that price per unit and variable cost per unit do not change over time.

I see. What is the rationale for the demand function p (lowercase) being synonymous with price per unit sold? Shouldn't price per unit be some other value that depends on demand AND supply?
 
I see. What is the rationale for the demand function p (lowercase) being synonymous with price per unit sold? Shouldn't price per unit be some other value that depends on demand AND supply?
You are mixing up two completely different models, which is hardly surprising given the utter and complete incompetence of this video.

There is a model of price setting in an economy with only two goods traded in fully competitive markets. This is the famous supply and demand model, where amount exchanged and the market price are simultaneously determined.

There is a different model about a single firm, the model of the firm. In that model, the firm faces known demand and cost curves and chooses what price to set and what quantity to sell based on maximizing its profit. Demand comes in through the revenue function, and supply comes in through the cost function. The two models can be integrated by assuming that the firm operates in perfectly competitive output and input markets and thus faces a horizontal demand curve (that is, it is a price taker rather than a price maker).

The problem posed does not actually have much to do with any standard economic model. It is based on the model of the firm in the context of an imperfect market, but has been altered to permit a discussion of related rates. And it has been all junked up with confusing notation and inconsistent vocabulary.
 
Okay, I'll avoid thinking too hard about this problem. Thanks for the advice.
 
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