What is the area of the shared region in the figure above that is bounded by the $x$-axis and the curve with the equation $y=x\sqrt{1-x^2}$?
This is the problem I was given. I assumed the answer was $0$ – the positive and negative areas should cancel each other out. That answer was incorrect. I then thought the answer might be $\frac13$ and they are only asking for the area above the $x$-axis. That was also incorrect. The answer that was given as correct was: $\frac23$. I assume that is because it is the area of the top – which is $\frac13$ and the area of the bottom also $\frac13$ – which makes $\frac23$. But, why is the answer not zero? Doesn't integration count area under the $x$-axis as negative? Does the wording of the question say otherwise?
Best Answer
In your question, it asks for the area of the shaded region, and area is always positive.
For integration, it is taught as the area between the curve and the $x$-axis. But actually not quite, because in the definition of integration we calculate "Area" as "NET area" (positive if above $x$-axis, and offset by those below $x$-axis).
Note that Area is absolute, but Net Area is relative