Question

Determine the average value of f(x,y) = e^x+y over the region R=[0,2]X[0,2].

Answer 1

Let $f(x,y)=e^{x+y}$ and $R=[0,2]\times [0,2]$ .

The average value of $f$ over $R$ is $f_\text{avg}=\frac{1}{\text{Area}_R}\iint _R f(x,y)dA$ .

$\text{Area}_R=2^2=4$

$\iint _R f(x,y)dA=\int\limits_0^2\int\limits_0^2e^{x+y}dxdy=\int\limits_0^2 \int\limits_0^2 e^{x}\cdot e^ydx dy= \int\limits_0^2e^{x}dx \cdot \int\limits_0^2e^ydy=e^x\big|^2_0\ \cdot \ e^y\big|_0^2=(e^2-1)^2$

Answer: $f_{\text{avg}}=\frac{(e^2-1)^2}{4}$ .