Riemann integral

Mathematics

1. (20 pts) Using the deÖnition of Riemann integral Önd the integrals of the following functions
a) f : [0; 1] ! R; f (x) = 
1 if x 2 [0; 1] n
1
2

1
if x =
1
2
b) f : [0; 1] ! R; f (x) =  1
n
if x =
1
n
; n 2 N
0 if x 2 [0; 1] n
 1
n
j n 2 N

2. (20 pts) Let f : [a; b] ! R be bounded and let M = sup ff (x) j x 2 [a; b]g, m = inf ff (x) j x 2 [a; b]g ;
Mf = sup fjf (x)j j x 2 [a; b]g, me = inf fjf (x)j j x 2 [a; b]g :
a) Show that, for all x; y 2 [a; b] ; jf (x)j
jf (y)j  M
m;
b) Show that Mf
me  M
m;
c) Show that, if f : [a; b] ! R is Riemann integrable then jfj : [a; b] ! R deÖned by
jfj(x) = jf (x)j
is Riemann integrable.
3. (20 pts) Let f; g : [a; b] ! R be bounded functions and let Mf = sup ff (x) j x 2 [a; b]g, mf =
inf ff (x) j x 2 [a; b]g ; Mg = sup ff (x) j x 2 [a; b]g, mg = inf ff (x) j x 2 [a; b]g :
a) Let M = sup ff (x) + g (x) j x 2 [a; b]g and m = inf ff (x) + g (x) j x 2 [a; b]g. Show that
M  Mf + Mg and m  mf + mg
b) Show that if f; g : [a; b] ! R are Riemann integrable then f+g : [a; b] ! R; (f + g) (x) = f (x)+g (x)
is Riemann integrable.
4. (20 pts) Let f : [a; b] ! R be an integrable function. DeÖne
f+(x) = (
f(x) if f(x)  0;
0 if f(x) < 0;
and f(x)
= (
0 if f(x)  0;
f(x)
if f(x) < 0:
Prove that f+, f
: [a; b] ! R are integrable and
Z b
a
f(x)dx =
Z b
a
f+(x)dx
Z b
a
f(x)dx:
5. (20 pts) Let f : [a; b] ! R be a Riemann integrable function such that
9 > 0 3 8x 2 [a; b]; f(x)  :
Show that the function g(x) = 1
f(x)
is also Riemann integrable on [a; b].
1
Extra Credit Problems
1. (30 pts) Let f : [0; 1] ! R be a di§erentiable function with f(0) = 0 and f
0
(x) 2 (0; 1) for every
x 2 (0; 1). Show that
Z 1
0
f(x)dx
2
>
Z 1
0
(f(x))3dx:
(Hint: Show that the function F(x) = R
x
0
f(t)dt
2

R x
0
(f(t))3dt is increasing.)
2. (30 pts) Compute the following improper integrals
Z 1
0
xe
x cos( x) dx; Z 1
0
xe
x sin( x) dx

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