91Ó°ÊÓ

In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{-1}^{1} \int_{x^{2}}^{1} f(x, y) d y d x $$

Find the average value of \(f(x, y)\) over the region \(R\) where Average value $$=\frac{1}{A} \int_{R} \int f(x, y) d A$$ and where \(A\) is the area of \(R\). \(f(x, y)=x\) \(R\) : rectangle with vertices (0,0),(4,0),(4,2),(0,2)

The population density of a city is approximated by the model \(f(x, y)=4000 e^{-0.01\left(x^{2}+y^{2}\right)}, x^{2}+y^{2} \leq 49,\) where \(x\) and \(y\) are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city.