MATH 296 PROBLEMS 12

IGOR KRIZ

Regular problems:

1. Let f : [−a, a] → ℝ. We say that f is even if f(−x) = f(x) and odd if f(−x) = −f(x). If f is Riemann-integrable on [0, a], prove that

(a) ∫_{−a}^{a} f(x)dx = 2 ∫_{0}^{a} f(x)dx is f is even

(b) ∫_{−a}^{a} f(x)dx = 0 is f is odd.

2. Integrate:

(a) ∫ \frac{sin(x)+cos(x)}{1+sin(x)}dx.

(b) ∫ \frac{dx}{1+\sqrt{x^{2}+2}}.

3. Use the definition of Riemann integral and “scaling” of partitions to prove that

∫_{a}^{b} f(x)dx = (b − a) ∫_{0}^{1} f(a + (b − a)x)dx.

4. The equations u = f(x, y, z), x = X(r, s, t), y = Y (r, s, t), and z = Z(r, s, t) define u as a function of r, s, and t, say, u = F (r, s, t). Use the chain rule to express the partial derivatives \frac{∂F}{∂r}, \frac{∂F}{∂s}, and \frac{∂F}{∂t} in terms of partial derivatives of f, X, Y and Z.

Challenge problems:

5. Consider the function f : ℝ → ℝ defined by

f(x) = \{ \begin{matrix}0 & if x is irrational \\ 1/n & if x = m/n where m is an integer, n is a posi- tive integer and there is no integer greater than 1 dividing both m, n \end{matrix}

Show that

∫_{0}^{1} f(x)dx = 0.

6. Suppose f : [a, b] → [0, +∞) is a bounded function (i.e. there is a constant M such that |f(x)| ≤ M for all x ∈ [a, b]) and suppose that there is a subset S ⊂ [a, b] such that

(1) f is continuous at every point outside of S, and

(2) For every ϵ > 0, S can be covered by finitely many intervals with total length < ϵ.

Prove that then f is Riemann-integrable.