Q:

Double iterated integration: what am I doing wrong?

The question is:
integrate $f(x,y) = 1$ on the region bounded by:
$$0 \le y \le -x^2 + 1$$
$$-1 \le x \le 0.5$$
I've turned this into the following integral:
$$\int_0^1 \int^{0.5}_{\sqrt{1-y}} 1 dx dy$$
When I evaluate this, I get $\frac{1}{6}$.  However, this is equivalent to the 1D integral:
$$\int_{-1}^{0.5} (-x^2 + 1)dx$$
which correctly evaluates to 1.125.
Clearly, my bounds are a problem, but I can't figure out where I went wrong.

A:

Try $$\int_0^{0.75} \int^{0.5}_{-\sqrt{1-y}} 1 dx dy $$ + $$\int_{0.75} ^1\int^{\sqrt{1-y}}_{-\sqrt{1-y}} 1 dx dy $$
Indeed, x=0.5 intersects the parabola in y=0.75

