Let be Riemann integrable on . Let be any real number, and let be the function given by . Prove that is Riemann integrable on and that . What is the geometric interpretation.

Proof. Let . Let be a partition of with . Let be such that . We know that . Now . So we just expand this out and see that ?

And the geometric interpretation is that we are increasing the each of the rectangle's height by . And we are doing this for rectangles.