Bounds of integration and uniform distribution (continuous)

Question

I have a probability problem. I'm a beginner, I hope my question isn't too stupid...

I would like to compute $\mathbb{P}(u^2 < 4v)$ with $u$ and $v$ following an uniform distribution (continuous) on $[-1,1]$.

I do $$\mathbb{P}(u^2 < 4v) = \mathbb{P}(u < 2 \sqrt{v}) = \iint_{\{u<2\sqrt{v}\}} \dfrac{1}{2} \dfrac{1}{2} \mathrm du \mathrm dv = \int_{-1}^1 \int_{-1}^{2\sqrt{v}} \dfrac{1}{4} \mathrm du \mathrm dv$$

But the result I find is unlikely. Could someone help me to find the good "integration bounds"? Sorry for my bad English...

Answer 1

Note that $u^{2}<v$ implies that $v>0$. Hence the range for 4v$ is 0 to 1, not -1 to 1.