I want to calculate the area of the following set $$B = \{ (x,y,z) \in R^3 | 4x^2+ y^2 - 1 \le z , 0 \le z \le 2y + 2 \}$$
This is part of an elliptic paraboloid (two planes cut it).
I thought that a correct integral to evaluate the area was
$$\int_C \int_0^{2y + 2} dz dxdy$$
Where $C$ is the set of all $y$, $x \in B$, so they must satisfy $4 x^2 + y^2 - 1 \le 2y + 2$. This means $$C = \{ (x,y) \in R^2 | 4x^2 +(y-1)^2 \le 4 \}$$
In this way I could pass to modified cylindrical coordinates and get a fairly quick solution. But I made a mistake with the limits of integration or the set $C$, where is the mistake in my reasoning?