Evaluate this Iterated Integral (Should Theoretically be Simple) I've been tasked with solving the following double integral:
$\int_0^1 $  $\int_{x}^{\sqrt{x}}({2x-y}) dydx$
I've repeated this question several times, and always end up with the answer 11/12. Integrating with respect to y first gives the integral $\int_0^1 $ $2x - \frac{x}{2} + \frac{x^2}{2}dx$, which should simplify to $\frac{24}{24} - \frac{6}{24} + \frac{4}{24}$. And yet the answer given in the book is $\frac{1}{20}$. Can someone point out what I'm doing wrong here? 

You may just write
$$
\begin{align}
\int_0^1\int_{x}^{\sqrt{x}}(2x-y) dydx&=\int_0^1\left[2xy-\frac{y^2}2\right]_{x}^{\sqrt{x}}dx\\\\
&=\int_0^1\left(\left(2x\sqrt{x}-\frac{x}2\right)-\left(2x^2-\frac{x^2}2\right)\right)dx\\\\
&=\int_0^1\left(2x^{3/2}  - \frac{x}{2} - 2x^2+ \frac{x^2}{2}\right)dx\\\\
&=\frac1{20}.
\end{align}
$$
