MATH 205 HOMEWORK #4

DUE DATE: THURSDAY, APRIL 25TH

1. Due before 6pm

The problems in this section may be handwritten; if they are typeset they may be turned in withthe second part of the problem set.Rudin: Chapter 9: 27, 29, 30.

Chapter 10: 2.

(1) Compute ∫ 1

0

x− 1

log xdx.

(Hint: ddyx

y = (log x)xy.)

2. Due before 11:59pm

The problems in this section must be typeset. Please email solutions, including both .tex and.pdf files, to zakh@math.uchicago.edu.Rudin: Chapter 9: 26.

(2) Construct a function f : [0, 1]× [0, 1] → R such that for all x0 ∈ [0, 1], the function g(y) =

f(x0, y) is integrable with∫ 10 g(y) dy = 0, but such that there exists a y0 ∈ [0, 1] such

that the function h(x) = f(x, y0) is not integrable. Thus∫ 10 (

∫ 10 f(x, y) dy) dx = 0, but∫ 1

0 (∫ 10 f(x, y) dx) dy is not well-defined. (Hint: see Rudin 10.2.)

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