Find measure of a set

Question

I need to calculate the measure of set $A=\{(x,y,z)\in R^3: 4x^2+y^2<4,x>0,x^2>z>0\}$In other words i need to calculate integral $\int_{A}1d\lambda_{3}$.Would be more than glad for checking what i did. Let:$$2x=t,y=y,z=z$$ jacobian is $\frac{1}{2}$ so we have: $$\int_{A}1d\lambda_{3}=\frac{1}{2}\int_{B}1d\lambda_{3}$$ where B is $\{(x,y,z)\in R^3: t^2+y^2<4,t>0,t^2>4z>0\}$ no i am using cylindrical transformation: $$x=r\cos\beta$$ $$y=r\sin\beta$$$$z=z$$ $-\pi<\beta<\pi$ and $r>0$ Our set is now $$C=\{r^2<4,r\cos\beta>0,r^2\cos^2\beta>4z>0\}=\{0<r<2,r\cos\beta>0,\frac{r^2\cos^2\beta}{4}>z>0\}$$ and jacobian is $r$ $$\int_{A}1d\lambda_{3}=\frac{1}{2}\int_{B}1d\lambda_{3}=\frac{1}{2}\int_{C}rd\lambda_{3}$$ using Fubbini: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{2}\int_{0}^{\frac{r^2\cos^2\beta}{4}}rdzdrd\beta$$ and to this moment is is correct?I hope it will look okay also as i have some problems with latex - it shows and disappears leaving plain text.