Find the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over the region R: square with vertices $( 0,0 ) , ( 2,0 ) , ( 2,2 ) , ( 0,2 )$ .
A) $\frac { 8 } { 3 }$
B) $\frac { 32 } { 3 }$
C) $\frac { 16 } { 5 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 8 } { 5 }$

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 $f(x, y) = x^2 + y^2$ over the square region with vertices $(0,0), (2,0), (2,2), (0,2)$, the area of R is $2 \times 2 = 4$. The double integral of $f(x, y)$ over R is $\int_{0}^{2} \int_{0}^{2} (x^2 + y^2) dy dx$, which evaluates to $\frac{8}{3}$ for each of $x^2$ and $y^2$ over the given limits, summing up to $\frac{16}{3}$. Dividing by the area, $4$, gives the average value as $\frac{16}{3} \div 4 = \frac{8}{3}$.

Find the Average Value Of f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2

Find the Average Value Of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$