Assume that b &gt

1. Homework 9 Let −∞ < a < b < ∞.

(1) Let α be any integer and f (x) = x^α for x ∈ R. Then f is continuous on R. We know f (x) is integrable over [a, b]. Assume that b > a > 0. Let n be any natural number and define h by hⁿ = b/a, i.e. h = pb/a. Setⁿ

xi = ahⁱ, 0 ≤ i ≤ n.

Then P = {x0= a < x1< · · · < xn= b} forms a partition of [a, b].

(a) Find the norm kP k of P.

(b) Consider the Riemann sum Fn=

n

X

i=1

x^α_i(xi− xi−1).

Show that F_n=

((b^α+1− a^α+1)_hα+hα−1^hα_+···+h+1 if α 6= −1

n^h−1_h if α = −1..

(c) Evaluate lim

n→∞F_n to obtain Z b

a

x^αdx for all α 6= −1. In the case of α = −1, show that Z b

a

1

xdx = lim

n→∞n ⁿ rb

a− 1

! . (2) Let e^xdenotes the exponential function

e^x=

∞

X

n=0

xⁿ

n!, x ∈ R.

Let us pretend that we can interchange the order of Z

andX

in other words, Z b

a

∞

X

n=0

xⁿ n!

! dx =

∞

X

n=0

1 n!

Z b a

xⁿdx.

Suppose that the above statement is true. Show that Z b

a

e^xdx = e^b− e^a.

(3) The function s(x) = sin x, x ∈ R is a continuous function on R. We know s(x) is Riemann integrable on [a, b]. (To be more precise, s|_[a,b] is Riemann integrable.) Let n be any natural number and h = b − a

n . Denote

x_i= a + ih, 0 ≤ i ≤ n.

Then P = {a = x₀ < x₁ < · · · < x_n = b} is a partition of [a, b]. The norm kP_nk of the partition is h. We know

R_n=

n

X

k=1

s(x_i)(x_i− x_i−1) =

n

X

k=1

h sin (a + kh)

is a Riemann sum of s with respect to P with mark C. Here Cn = {x1< · · · < xn}.

1

2

(a) Use the identity −2 sin A sin B = (cos(A + B) − cos(A − B)) to show that Rn= h

2 sin^h₂



cos(a +h

2) − cos(b −h 2)

 .

(Hint: Multiply Rn by 2 sinh 2.) (b) Evaluate lim

n→∞R_n to obtain Z b

a

sin xdx.

(c) Similarly, evaluate Z b

a

sin λxdx, λ ∈ R.

(4) Consider the function c(x) = cos x, x ∈ R. Then c is Riemann integrable on [a, b]. Let Sn =

n

X

k=1

h cos(a + kh).

(a) Evaluate lim

n→∞Snto obtain Z b

a

cos xdx. (Hint: Use 2 cos A sin B = sin(A + B) − sin(A − B).)

(b) Similarly, evaluate Z b

a

cos λxdx, λ ∈ R.

(5) Let λ, µ ∈ R. Using the previous exercises and the following trigonometric identities 2 sin A cos B = sin(A + B) + sin(A − B)

2 cos A cos B = cos(A + B) + cos(A − B) 2 sin A sin B = cos(A − B) − cos(A + B).

to evaluate the following definite integrals.

(a) Z b

a

sin λx sin µxdx.

(b) Z b

a

sin λx cos µxdx.

(c) Z b

a

cos λx cos µxdx.

(6) Show that any a bounded monotone function on [a, b] is Riemann integrable over [a, b].

(7) Use the property of integrals to prove the following inequalities.

(a) 1 10√

2 ≤ Z 1

0

x⁹

√1 + xdx ≤ 1 10. (b) 11

24 ≤ Z ¹₂

0

p1 − x²dx ≤ 11 12√

3. (8) Evaluate

(a) lim

n→∞

Z 1 0

xⁿ 1 +√

xdx.

(b) lim

n→∞

Z ^π₂

0

sinⁿxdx.

(9) Suppose f ∈ C[0, 1] and Z 1

0

f (x)x¹⁰dx = 5, Z 1

0

f (x)

 x⁶−x⁵

6 + 1



dx = −6.

Prove that f (x) = 0 has a solution in [0, 1].

(10) Suppose C₀, C₁, · · · , C_n are real numbers such that C0+C1

2 + · · · +Cn−1

n + Cn

n + 1 = 0.

3

Prove that the polynomial equation

C0+ C1x + · · · + Cn−1xⁿ⁻¹+ Cnxⁿ= 0 has at least one real root between 0 and 1.

(11) Let f (x) be a nonnegative continuous function on [a, b], i.e. f ∈ C[a, b] and f (x) ≥ 0 for all x ∈ [a, b]. Show that if

Z b a

f (x)dx = 0,

then f is the zero function on [a, b], i.e. f (x) = 0 for all a ≤ x ≤ b. Use the following steps.

Suppose that f (x0) > 0 for some x0∈ [a, b].

(a) Prove the following statements.

(i) If x0= a, then there exists δ > 0 and m > 0 such that f (x) > m for a ≤ x ≤ a+δ.

(ii) If x₀= b, then there exists δ > 0 and m > 0 such that f (x) > m for b − δ ≤ x ≤ b.

(iii) If a < x₀ < b, then there exists δ > 0 and m > 0 such that f (x) > m for x₀− δ ≤ x ≤ x₀+ δ.

(b) Use (a) to prove the following statements.

(i) If x₀= a, or x₀= b then Z b

a

f (x)dx ≥ mδ.

(ii) If a < x₀< b,

Z b a

f (x)dx ≥ 2mδ.

(12) Let f (x), g(x) ∈ C[a, b].

(a) Prove the following Cauchy-Schwarz inequality Z b

a

f (x)g(x)dx

!²

≤ Z b

a

f (x)²dx Z b

a

g(x)²dx.

Hint: Use the following steps:

(i) Assume that f (x) is not the zero function in C[a, b]. Define F (t) =

Z b a

(tf (x) − g(x))²dx, t ∈ R.

Find A, B, C such that F (t) = At²− 2Bt + C.

(ii) Show that A > 0.

(iii) Show that F (t) ≥ 0 for all t ∈ R.

(iv) Using (b), show that B²≤ AC.

(b) Show that the equality holds if and only if g(x) = λf (x) on [a, b] for some constant λ.

Hint: use the previous exercise.

(13) (Generalized mean value theorem) Let f (x) ∈ C[a, b] and g ∈ R[a, b]. Suppose that g(x) > 0 on [a, b]. Prove that there exists ξ ∈ [a, b] such that

Z b a

f (x)g(x)dx = f (ξ) Z b

a

g(x)dx.