Solution
= Solution
The <domain of integration> is the upper half of the <ellipse> $x^2+4y^2\leq1$. Reversing the order makes $-1\leq x\leq1$ and $0\leq y\leq\tfrac12\sqrt{1-x^2}$. Thus
$$
\boxed{\int_{-1}^1\int_0^{\sqrt{1-x^2}/2}y\,dy\,dx=\frac18\int_{-1}^1(1-x^2)dx=\frac16.}
$$
The inner antiderivative is $y^2/2$, so the square of the upper limit removes the square root.