I'm self-learning multivariable calculus and am using double integration to compute the average value of $f(x,y)$ over some region. I'm trying to solve the following simple problem using two different approaches, with double integration and with basic intuition only, but am getting slightly different results and am wondering why (and feeling quite stupid as I'm sure I'm missing something very simple).
The problem is as follows:
A company's daily profit when selling two products is given by: $$P(x,y)=192x + 576y - 2xy$$ where $x$ and $y$ is the number of each product sold. Estimate the average daily profit if $x$ varies between 40 and 50 units, and $y$ varies between 45 and 50 units.
Using double integration, the Mathematica-verified solution is: $$\text{avg} = \frac{1}{50}\cdot \iint\limits_{40\;45}^{50\;50} P(x,y)\;dy\;dx = 31725$$
Now if I forget about integration, my intuitive solution would look like this:
- compute the minimal possible profit on a given day: $$P(40,45)=30000$$
- compute the maximal possible profit on a given day: $$P(50,50)=33400$$
- compute the average of these two values (sum and divide with 2), which yields 31700.
While the results are close, 31700 is clearly not 31725, so something must be wrong with my second solution, i.e. with my intuition. What am I missing?
The average of $10$, $99$, and $101$ is $\dfrac{10+99+101} 3 = 70$, and that is not the same as the average of the smallest and largest of these numbers, which is $\dfrac{10+101} 2 = 55.5$. That is because $99$ is a lot bigger than the average of the smallest and largest ones. It is not generally true that the average of the numbers in a list is the same as the average of the smallest and largest one. (However, it is true of arithmetic sequences.)