MATH 395 PROBLEMS 12

IGOR KRIZ

Regular problems:

1. Recall that we say that ∫ f exists if the function f is Lebesgue-integrable, and that ∫ f converges if it exists and is finite. For what values of a do the following integrals converge:

∫_{0}^{1} x^{a}dx, ∫_{1}^{∞} x^{a}dx, ∫_{0}^{∞} x^{a}dx?

2. Compute

∫_{[0,1]×[0,1]} max(x, y)dxdy.

[Use Fubini’s theorem, but not mechanically.]

3. Decide whether the following integrals exist and/or converge: [use bounds by integrals you know]

(a) ∫_{2}^{∞} \frac{dx}{\sqrt{x^{4}−1}}

(b) ∫_{0}^{π} \frac{sin(x)}{x}dx

(c) ∫ \frac{dx}{x^{2}+x}.

4. Let f : [0, ∞) → ℝ be defined by f(0) = 0 and

f(x) = \frac{(−1)^{n}}{n} for x ∈ (n − 1, n].

Prove that the Lebesgue integral ∫ f does not exist, but the improper Riemann integral

lim_{t→∞} ∫_{0}^{t} f(x)dx

exists. Is there such an example with f ≥ 0? [In proving the non-existence, use the fact that an g ∈ Z^{inc} such that g ≥ f is bounded below by a function from Z, and hence is non-negative outside of a bounded interval; similarly, g ∈ Z^{dec} such that g ≤ f is non-positive outside of a bounded interval; conclude that the upper integral of f is +∞, the lower integral is −∞.] Challenge problems:

5. Define the ζ-function ζ : (1, ∞) → ℝ by

ζ(x) = ∑_{n=1}^{∞} n^{x}.

Does the Lebesgue integral

∫_{1}^{2} ζ(x)dx

converge?