\begin{align} D &= \left\{ (r \cos \theta, y \sin \theta) \middle| 1 \leq r \leq e, \ 0 \leq \theta \lt 2 \pi \right\} \end{align} と書けるので、次のように計算できる： \begin{align} \iint_D \log \left( x^4 + 2x^2y^2 + y^4 \right) dxdy &= \int_0^{2 \pi} d \theta \int_1^e dr \ r \log \left( r^4 \cos^4 \theta + 2 r^4 \sin^2 \theta \cos^2 \theta + r^4 \sin^4 \theta \right) \\ &= \int_0^{2 \pi} d \theta \int_1^e dr \ r \cdot 4 \log r \\ &= 8 \pi \int_1^e dr \ r \log r \\ &= 4 \pi \left[ r^2 \log r \right]_1^e - 4 \pi \int_1^e dr \ r \\ &= 4 \pi e^2 - 2 \pi \left[ r^2 \right]_1^e \\ &= 4 \pi e^2 - 2 \pi e^2 + 2 \pi \\ &= 2 \pi \left( e^2 + 1 \right) . \end{align}