Double Integrals — Question 8

Question 8 – Solution

Strategy. The average of an affine function over a rectangle equals its value at that rectangle’s center; use the three given centers to reconstruct the coefficients.

Step 1: Average equations The relevant centers are $(1/2,1/2)$, $(3/4,1/2)$, and $(1/2,3/4)$. Hence $$\begin{align*} a+\frac b2+\frac c2&=4,\\ a+\frac{3b}{4}+\frac c2&=5,\\ a+\frac b2+\frac{3c}{4}&=3. \end{align*}$$ The center-value rule follows from the same reflection pairing used for any affine graph over a rectangle.

Step 2: Reconstruct Subtracting the first equation from the second gives $b/4=1$, so $b=4$. Subtracting the first from the third gives $c/4=-1$, so $c=-4$. The first equation then gives $a=4$. Therefore $$\boxed{f(x,y)=4+4x-4y}.$$

Step 3: New prediction The lower-left quarter has center $(1/4,1/4)$ and area $1/4$. Its average is $$f\left(\frac 14,\frac 14\right)=4+1-1=4,$$ so $$\boxed{\iint_{[0,1/2]^2}f\,dA=4\left(\frac 14\right)=1}.$$ The equal $x$- and $y$-center coordinates cancel the opposite slopes, which checks the predicted average.