Area enclosed by a curve and the coordinate axes

Question

While finding out the area enclosed by curves and the $2$-D coordinate axes, I found something strange. Let's assume we have a parabola $y^2 = 16x$. Now the y coordinate would be given by $4x^{1/2}$. Integrating this expression with limits from, say, a to 0, we get a certain value we call the area. But, why do we always obtain the area in the first quadrant? The $x$ coordinate is positive in the fourth quadrant too, so shouldn't the area we obtain, include that in the fourth quadrant as well? Mathematically, I don't see a particular reason for this. Any help in understanding this would be great.Thanks in advance.

Answer 1

Yes, "the area enclosed by" these things is ambiguous. If you're asked to find it, the questioner should be sure to be clear and specific, like "the finite region above the $x$-axis and bounded by $y^2=16x$ and $x=a$ ($a>0$)". Or if a question doesn't mention the axes and asks for "the area of the region enclosed by $y^2=16x$ and $x=a$ ($a>0$)", then this region would be in both the first and fourth quadrants.