Double Integration Inequality I've been trying to work out the following. Could anyone please show me how to do this?

Let $D$ be the domain bounded by $y=x^2+1$ and $y=2$. Prove the inequality
$$\frac{4}{3}\le \iint_D(x^2+y^2)\,\mathrm{d}A\le\frac{20}{3}.$$

Thank you.

First notice that in $D$ we have $1 \le x^2 + y^2 \le 5$. This implies $$ |D| \le \iint_Dx^2 + y^2 \le 5|D|$$
Now we only need to find $|D|$ ( = area of $D$), which is of course equal to $$4 - \int_{-1}^1(x^2 + 1) = 4 - 2\int_0^1(x^2 + 1) = 4 - \frac{2}{3} - 2 = \frac{4}{3}$$
