Double Integrals — Question 6

Question 6 – Solution

Strategy. Pair each point $(x,y)$ with its central reflection $(-x,-y)$ and average the two function values.

Step 1: Pairing Let $h(x,y)=x^3+xy^2+2y$. Then $$h(-x,-y)=-x^3-xy^2-2y=-h(x,y).$$ Thus the contributions of $h$ at every reflected pair cancel. Equivalently, $$f(x,y)+f(-x,-y)=10.$$

Step 2: Integral and average Central reflection maps $R$ onto itself without changing area. Therefore the paired average of $f$ is $5$ everywhere, and $$\iint_Rf\,dA=5\operatorname{area}(R)=5(4)(2)=\boxed{40}.$$ Dividing by the area $8$ gives $$\boxed{f_{\mathrm{avg}}=5}.$$

Step 3: Symmetry check The required property is $$(x,y)\in R\quad\Longleftrightarrow\quad(-x,-y)\in R.$$ Symmetry in only one coordinate would not automatically cancel the entire $h$: $xy^2$ is odd in $x$, while $2y$ is odd in $y$. Central symmetry reverses both coordinates at once and cancels all three terms.