Setting domain for double integral

Question

I am given the following integral $\iint (x^2+y^2)e^{x^2+y^2} dxdy$ and the domain D={(x,y)| $1\le x^2+y^2 \le 4$}

And I am asked to compute the integral, I think I know how to compute it, but I am having trouble setting it up.

Answer 1

In this case you should use polarcoordinates as follow:

Let us consider the change of coordinates given by: $$ x=r \cos \theta ,\, y=r \sin \theta $$ then the Jacobian (intuitively this is the small deformation in the space that occur by our change of variable) of this transformation is given by $$\begin{vmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{vmatrix}=r$$

Now for change the elements of our integral let us consider this useful fact

$$x^2+y^2=r^2 \cos^2 \theta + r^2 \sin^2 \theta=r^2(\cos^2 \theta + \sin^2 \theta)=r^2$$ that help us to change and remplace in the integral the expression $x^2+y^2$.

Now we are ready for change our integral (which is only switch all in terms of $r$ and $\theta$ with the respectively small change in the deformation The Jacobian and our new region of integration) to a new integral which is easy of calculate, for this task let us calculate how looks our new domain $W$.

Since $D$ is given by $D=\lbrace (x,y)\in \mathbb{R}^2\mid 1 \leq x^2+y^2 \leq 4\rbrace$ then by our change of variable it stay like $$W=\lbrace (r,\theta)\mid 1\leq r^2\leq 4 \rbrace $$

Now our integral is $$\iint_W r^2 e^{r^2} r d \theta d r= \int_1^2 \int_0^{2\pi} r^3 e^{r^2} d \theta dr=2\pi \int_{1}^{2}r^3 e^{r^2}dr= 2\pi \frac{3}{2}e^4=3\pi e^4$$