The integral of f: R -> R represents area, not height. Viewed as a sum of rectangles A1 + ... + An, with Ai = hi*wi, the height (h) or width (w) can be larger than 1, but their product cannot be. That is, hi > 1 OR wi > 1, for some i, but hi*wi = Ai < 1, for all i.
An example: f(x) = 2, for 0 < x < 1/2. This is a valid probability density (uniform with parms. a = 0, b = 1/2) and f(x) > 1 for any x.