Q:

Evaluate double integral $\iint_A \log(y)\,dx\,dy$

Calculate the double integral
$$\iint_A \log(y)\,dx\,dy$$
Where 
$$ A = \{(x,y)\in R^2 : 0 \leq x \leq 1, \sqrt{x}\leq y \leq \sqrt[3]{x}\}$$
Help! 

A:

Well, I would invert the inequalities to $y^3\le x\le y^2$ and $0\le y\le 1$, and set up your iterated integral as follows:
\begin{align*}
\iint_A\ln(y)\,dx\,dy
&=\int_0^1\ln(y)\int_{y^3}^{y^2}dx\,dy\\
&=\int_0^1\ln(y)\left(y^2-y^3\right)dy\\
&=\left[\frac{y^4}{16}-\frac{1}{4} y^4 \ln (y)-\frac{y^3}{9}+\frac{1}{3} y^3 \ln(y)\right]_0^1.
\end{align*}
Recall that these are limits, so you might have to use L'Hospital's Rule to evaluate on the lower end.

