Q:

Finding Area between the Y-Axis and a Parabola by integration.

For a National Board Exam Review:

Find the area bounded by the parabola $x^2 = 4y$ and $y=4$.

Answer is $21.33$
I could do the $X$-axis; but I want to try to integrate through the $Y$-axis so if in the real exam something comes up that really requires me to integrate vertically then I would know how. This was supposed to be straightforward, so I simply put in my calculator: 
$${ \int^{4}_{0} { \sqrt{ 4y } \mathrm dy } = 10.66667}$$ 
What am I doing wrong? I have a feeling I am getting the area on the opposite side...

A:

Equating$y=\frac{x^2}{4}$ and $y=4$ we get $x=4$ and $x=-4$
you have to integrate $y$ from $-4$ to $4$ 
$$\int_{-4}^{4} \frac{x^2}{4}dx=\frac{x^3}{12}\mid_{-4}^{4}$$ and you will get the answer.In integrating with respect to $y$ you only computed the half of the area.

