I just tried to visually solve the integral of $\sin(x)$ from $0$ to $\frac{\pi}{2}$ by considering the unit circle.
I tried to sketch my idea. The link is below: https://drive.google.com/file/d/0B-uFKRww3qhWRk5oVGRHRENVY28/view?usp=sharing
If one lays a Cartesian coordinate system with its origin coinciding with the center of the unit circle, then the y-coordinate can be thought of as representing the value of $\sin(x)$ at a given angle.
If I would add all the values from $2$ to $\frac{\pi}{2}$, this is to my understanding the same as visually drawing a line for each x by gradually taking a little step $dx$. This corresponds to my understanding of an integral as a Riemann sum. The sum of all the "lines" is the area, namely $\frac{1}{4}$ of the unit circles area. Ergo $\frac{1}{4}$.
As I found out analytically the integral is not $\frac{\pi}{4}$. Wherein lies my error?
I am very grateful for your help in advance!