Calculate $\int\int_E e^{5x^2+2xy+y^2}dA$

Question

How to calculate $$\int\int_E e^{5x^2+2xy+y^2}dA,$$ where $E=\{(x,y)\mid 5x^2+2xy+y^2\leq 1\}$? I know I have to use the change-of-variable formula by first finding the change-of-variable function $\Psi$. However, what's the function should I use? And how to deal with the annoying $xy$-term?

Answer 1

Assume that the quadratic form $q(x,y)=ax^2+2bxy+cy^2$, associated to the matrix $Q=\begin{pmatrix}a & b \\ b & c \end{pmatrix}$ is positive definite, i.e. $a>0$ and $ac>b^2$ (Sylvester's criterion). Let $$E_q=\{(x,y)\in\mathbb{R}^2: q(x,y)\leq 1\}.$$ By the spectral theorem, $$ \iint_{E_q}e^{q(x,y)}\,dx\,dy = \iint_{\lambda_1 x^2+\lambda_2 y^2\leq 1}e^{\lambda_1 x^2+\lambda_2 y^2}\,dx\,dy $$ where $\lambda_1,\lambda_2$ are the eigenvalues of $Q$. By straightforward substitutions, the RHS equals $$ \frac{1}{\sqrt{\lambda_1 \lambda_2}}\iint_{X^2+Y^2\leq 1}e^{X^2+Y^2}\,dX\,dY=\frac{1}{\sqrt{\det Q}}\int_{0}^{2\pi}\int_{0}^{1}\rho e^{\rho^2}\,d\rho\,d\theta=\color{red}{\frac{\pi}{\sqrt{ac-b^2}}}. $$ Can you see what happens by considering $a=5$ and $b=c=1$?