Find the average value of the function f over the given region.
-$f ( x , y ) = \frac { 1 } { x y }$ over the square $1 \leq x \leq 3,1 \leq y \leq 3$.
A) $\left( \frac { \ln 3 } { 2 } \right) ^ { 2 }$
B) $\left( \frac { \ln 3 } { 3 } \right) ^ { 2 }$
C) $\frac { \ln 3 } { 9 }$
D) $\frac { \ln 3 } { 4 }$

Question

Quizplus · Accepted Answer

The average value of a function $f(x, y)$ over a region R is given by $\frac{1}{\text{Area of R}} \int \int_{R} f(x, y) dA$. For the given function $f(x, y) = \frac{1}{xy}$ over the square $1 \leq x \leq 3, 1 \leq y \leq 3$, the area of the region is $2 \times 2 = 4$. The double integral becomes $\frac{1}{4} \int_{1}^{3} \int_{1}^{3} \frac{1}{xy} dy dx$. Solving this integral gives $\left( \frac{\ln 3}{2} \right)^2$, which matches choice A.

Find the Average Value of the Function F Over the Given