The Riemann sum does two things for us.

1. It gives us a method for computing an approximation of an integral.
2. It gives us a way to make that approximation "arbitrarily close" to the exact value of the integral.

In the above applet we will see how that process works. You can move sliders and check boxes, but also, you can click and drag the function, and even define your own functions using the text box below the applet.