I am trying to find the time for different paths for a traversing ball between two points under the influence of gravity, but I am getting strange results for a reason I don't understand, I feel like I am missing something.
Time is equal to: $$ T=\sqrt{\frac{1}{2g}}\int_0^1\sqrt{\frac{1+(y')^2}{y}}dx $$ where $y$ is the functions I am trying to test.
My two points are $(0,1)$ and $(1,0)$, the former one being the starting point. I am trying to use a circle as one of my tests: $$ (x - h)^2 + (y - k)^2 = r^2 $$ Expressing $y$ as a function of $x$: $$ y=k\pm \sqrt{r^2-(x-h)^2} $$ And finally: $$ y=1-\sqrt{1^2-(x-1)^2}. $$ So I input the above and its derivative into my time equation: $$ T=\sqrt{\frac{1}{2g}}\int_0^1\sqrt{\frac{1+\bigg(\frac{x-1}{\sqrt{1-(x-1)^{2}}}\bigg)^2}{1-\sqrt{1^2-(x-1)^2}}}dx $$ But when putting this into Desmos it gives me $6.060$, which seems way longer than expected given that a straight line only takes $0.63$ seconds.
Am I doing something wrong?