Elementary Analysis Math 140B—Winter 2007Homework answers—Assignment 26; March 19, 2007

Exercise 32.6

Let f be a bounded function on [a, b]. Suppose there exist sequences (Un) and (Ln) of upper andlower Darboux sums for f such that lim(Un − Ln) = 0. Show f is integrable and

∫ b

af = lim Un =

limLn.

Solution: We are given that Un = U(f, Pn) and Ln = L(f,Qn) for certain partitions Pn andQn of [a, b]. Since

U(f, Pn ∪Qn)− L(f, Pn ∪Qn) ≤ Un − Ln → 0

it follows that for any ε > 0, there is a partition Pε = Pn ∪Qn for some n such that

U(f, Pε)− L(f, Pε) < ε.

Hence f is integrable on [a, b].

From Ln ≤ L(f) = U(f) =∫ b

af ≤ Un we have 0 ≤

∫ b

af −Ln ≤ Un−Ln and Ln−Un ≤

∫ b

af ≤ 0

and therefore limn Ln = limn Un =∫ b

af .

Exercise 33.8

Let f and g be integrable functions on [a, b]

(a) Show that fg is integrable on [a, b].

Solution: Since 4fg = (f + g)2 − (f − g)2 and Exercise 33.7 states that the square of anintegrable function is integrable, using linearity (Theorem 33.3) it follows that fg is integrable.

(b) Show that max(f, g) and min(f, g) are integrable on [a, b].

Solution: Since min(f, g) = (f +g)/2−|f−g|/2 and max(f, g) = −max(−f,−g), using The-orem 33.5 and linearity (Theorem 33.3), it follows that max(f, g) and min(f, g) are integrable.

Exercise 34.2

Calculate

(a) limx→01x

∫ x

0et2 dt

Solution: Let F (x) =∫ x

0et2 dt. Then limx→0

1x

∫ x

0et2 dt = limx→0

F (x)x = F ′(0) = ex2 |x=0 =

1.

(b) limh→01h

∫ 3+h

3et2 dt

Solution: Let g(x) =∫ x

3et2 dt so that g(3) = 0 and g′(x) = ex2

. Then limh→01h

∫ 3+h

3et2 dt =

limh→0g(3+h)−g(3)

h = g′(3) = e9.

Exercise 34.9

Use Example 3 to show∫ 1/2

0sin−1 x dx = π/12 +

√3/2− 1

Solution: Take a = 0, b = π/6 and g(x) = sin x in Example 3. Then∫ π/6

0

sinx dx +∫ 1/2

0

sin−1 x dx =π

6· 12

=π

12

and ∫ π/6

0

sinx dx = − cos x |π/60 = −

√3

2+ 1.

1