How am I supposed to approach this problem?

Question

Problem: The Cobb-Douglas production function for an automobile manufacturer is $$f(x,y)=100x^{.6}y^{.4} $$ Where x is the number of units of labor and y is the number of units of capital. Estimate the average production level if the number of units of labor x varies between 200 and 240 and the number of units of capital y varies between 300 and 330.

Answer 1

The average value of a function of two variables $f(x,y)$ over a region $R$ is given by: $$ \frac{1}{A(R)}\int_R f(x,y)dA $$

in your case the region is the rectangle defined by: $200<x<240$ and $300<y<330$; so the area is $A(R)= 1200$ and the integral becomes $$ \frac{1}{1200}\int_{300}^{330}\int_{200}^{240}100\; x^{0.6}y^{0.4} dx dy $$