Sample Integration Questions Math 127B. Winter, 2005

1. Give an example of a function f : [0, 1] → ℝ such that f^{2} is Riemann integrable, but f is not.

2. Suppose that f : [a, b] → ℝ is a bounded, Riemann integrable function. Define F : [a, b] → ℝ by

F (x) = ∫_{a}^{x} f(t) dt.

Prove that there exists a constant M such that

|F (x) − F (y)| ≤ M|x − y| for all x, y ∈ [a, b].

Is F necessarily differentiable in (a, b)?

3. Suppose that g : ℝ → ℝ is continuous. Define f : ℝ → ℝ by

f(x) = ∫_{0}^{x}(x − t)g(t) dt.

Prove that f satisfies the following equations:

f′′(x) = g(x), f(0) = f′(0) = 0.

4. Define the improper integral

∫_{0}^{∞} \frac{sin x}{x} dx

as a limit of proper integrals, and prove that it converges. Hint. Use integration by parts to show that the proper integrals form a Cauchy sequence.

5. Suppose that

F (x) = \{ \begin{matrix}x^{2} & for 0 ≤ x < 2, \\ x^{3} & for 2 ≤ x ≤ 3. \end{matrix}

Evaluate the Riemann-Stieltjes integral

∫_{0}^{3} xdF (x),

briefly justifying your computations.