I have chosen to investigate the fact that cycloid is a quicker path than the straight line for my HL Maths IA. I did my own experiment and was advised to only explain up to 'timing the fall' of the brachistochrone problem by my teacher. I did and got the equation:
$$T=\frac{1}{\sqrt{2}g} \int \frac{\sqrt{1+\left(y'\right)^{2}}}{\sqrt{y}} dx$$
However at this point my teacher wants me to compare values I got from the experiment for both straight line and cycloid to prove that the formula works and that the cycloid is faster. Where $y$ is the height, $g$ is the gravity, and $T$ is time. The height from the experiment was 18.8 cm but I don't know how to put that into the equation. Please could someone help me evaluating that equation?