1. Homework 1 (1) Let Ω be the following plane region
Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x3}.
(a) Sketch the region Ω.
(b) Prove that Ω has an area and computes its area, i.e. show that A+(Ω) = A−(Ω) and find A(Ω).
Hint: you need the formula
n
X
i=1
i3= n(n + 1) 2
2
.
(2) Let a, b be distinct real numbers such that a < b. f : [0, 1] → R be the function f (x) =
(a if x is a rational number in [0, 1];
b if x is an irrational number in [0, 1].
(a) Let P be any partition. Prove that U (f, P ) = b and L(f, P ) = a.
(b) Prove that Z 1
0
f (x)dx = b and Z 1
0
f (x)dx = a.
(c) Is f Riemann integrable?
(3) Let f : [0, 1] → R be the function f (x) = x2. For each natural number n, let Pn be the partition of [0, 1] :
Pn= {x0= 0 < x1= 1
n < · · · < xi = i
n < · · · < xn= 1}.
Let
Mi= sup
x∈[xi−1,xi]
f (x), mi= inf
x∈[xi−1,xi]f (x).
(a) Find Miand mi for all 1 ≤ i ≤ n.
(b) Prove that
U (f, Pn) =(n + 1)(2n + 1)
6n2 , L(f, Pn) = (n − 1)(2n − 1)
6n2 .
(c) Use the fact that (1.1)
Z b a
f (x)dx = sup
P
L(f, P ), Z b
a
f (x)dx = inf
P U (f, P ), and the fact that
Z b a
f (x)dx ≤ Z b
a
f (x)dx to show that
(n − 1)(2n − 1)
6n2 ≤
Z 1 0
x2dx ≤ Z 1
0
x2dx ≤ (n + 1)(2n + 1)
6n2 .
Remark. In (1.1), P runs through all partitions P of [a, b].
(d) Prove that f is Riemann integrable and compute Z 1
0
x2dx by Problem (3c).
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