Elementary Functions and Calculus III Math 133 (Sec 42), Spring 2005 Problem Set 5 (final question)

For a ∈ R, consider the function f_{a} : R → R defined by

f_{a}(t) = \{ \begin{matrix}1, if t ≥ a; \\ −1, if t < a. \end{matrix}

1. Show that the improper integral ∫_{−∞}^{+∞} f_{a}(x) dx diverges.

2. Show that lim_{b→+∞} ∫_{−b}^{b} f_{a}(x) dx converges (i.e. exists and is finite). What is its value?

3. Write a short paragraph explaining why you think we do not choose to define the improper integral ∫_{−∞}^{+∞} f(x) dx, for any function f, as in part 2.