Say we have $y = x^{0.5}$ and we want to find the area under the curve between the values of $0$ and $1$.
$\int_0^1 y dx = \left(\frac{2}{3} x^{1.5}\right)_0^1$
Now $x^{1.5} = x\cdot x^{0.5}$
If we plot the curve it is obvious that the area is positive, but what's to say, in the equation $\frac{2}{3} x^{1.5}$ that the answer is positive, a possible root of $x = -1$ so the area $= \pm \frac{2}{3}$
Is there an explanation or do we simply use the positive root because we know it has to be the positive root? Is there any case when you instead take the negative root?
Best Answer
Because most people in the world do not know that $\sqrt{x^2}=|x|$.
In another hand, if we write $x^{0.5}$ then by definition $x>0$.