Demand function and calculus, relation?

Question

i havent taken any econometrics course or anything

and im wondering if you have a demand function,

im not sure what kind, lets say demand as a function of price....

ive heard that theres methods of integration to help you find new information, what happens (a function of what) when you integrate this?

Answer 1

If you have a demand function:  Let's say it's Q (quantity)  = 10 - P (price)  (this is actually an inverse demand function because when you graph it, you will graph it as P = 10 - Q.  P is always on the y-axis and Q is on the x-axis).  So, demand is P = 10 - Q.  This means as the price increases, the quantity demanded is less.

I will assume you know what it means when I say integrate this equation from 0 to 10.  

Integrating from 0 to 10 gives [10*Q - (Q^2)/2] from 0 to 10.  

This gives [10*10-(10^2)/2] - [10*0 - (0^2)/2]

The second part is two, so the whole thing is 100- (100/2) = 50.  

When you integrate, in general, the result is the "area under the curve."  So, you have just integrated the area under the demand curve.  This is equivalent to some amount of the x-axis times some amount of the y-axis.  But, because the x-axis is Q and the y-axis is P, the area under the curve will be Q*P.  

The quantity you sell times the price gives you the amount of money you make (total revenue).  

Therefore, in my example, total revenue = 50, if you sell 10 (because you integrated from 0 to 10).  Generally, a firm sets a price, though.  In this example, you are doing all of the area under a curve, so we are implicitly assuming that you are charging every person the price that they demand the good at (something like a perfect price-discriminating monopoly).  

If you want to find total revenue for a perfect price discriminator at any quantity, just do the integral of the demand curve from 0 to the quantity you're interested in.  

Generally, when a firm sets a price (5, for example), you just take that price times the quantity demanded at the price (using inverse demand, this is Q = 10 - P, so Q = 5), so total revenue would be P*Q = 5*5 = 25, in this example.