Question

Determine the area of region by 𝒚 = x^2 + 4x and the y - axis.

Answer 1

There is no such region bounded by the y-axis, but there is a region bounded by the x-axis: The area of the small loop which dips below the axis. Since the curve intersects the axis at (-4,0) and (0,0), you want the area below y=0 and above the parabola. That would just be
∫[-4,0] (0 - (x^2+4x)) dx
or, using the symmetry of the region about the line x = -2, and cleaning up the minus signs,
-2∫[-2,0] (x^2 + 4x) dx