220 CHAPTER 9. INTEGRATION

the special value between a and a+1, then the area of A is equal to area of B. Whyshould x < a+ 1

2 ? Note that if A is the area of A and B the area of B,

f (x) ·1 =∫ x

af (t)dt +

∫ b

xf (t)dt = ( f (x)(x−a)−A)+( f (x)(a+1− x))+B

= f (x)+B−A

AB

xa a+1

Now use this to obtain the inequality 9.9.

32. Suppose f and g are continuous functions on [a,b] and that g(x) ̸= 0 on (a,b) .Show there exists c ∈ (a,b) such that f (c)

∫ ba g(x) dx =

∫ ba f (x)g(x) dx. Hint: De-

fine F (x) ≡∫ x

a f (t)g(t) dt and let G(x) ≡∫ x

a g(t) dt. Then use the Cauchy meanvalue theorem on these two functions.

33. Consider the function

f (x)≡{

sin( 1

x

)if x ̸= 0

0 if x = 0.

Is f Riemann integrable on [0,1]? Explain why or why not.

34. The Riemann integral is only defined for bounded functions on bounded intervals.When f is Riemann integrable on [a,R] for each R > a define an “improper” inte-gral as follows.

∫∞

a f (t)dt ≡ limR→∞

∫ Ra f (t)dt whenever this limit exists. Show∫

∞

0sinx

x dx exists. Here the integrand is defined to equal 1 when x = 0, not that thismatters.

35. Show∫

∞

0 sin(t2)

dt exists.

36. The most important of all differential equations is the first order linear equation,y′+ p(t)y = f (t) where p, f are continuous. Show the solution to the initial valueproblem consisting of this equation and the initial condition, y(a) = ya is given byy(t) = e−P(t)ya+e−P(t) ∫ t

a eP(s) f (s) ds, where P(t) =∫ t

a p(s) ds. Give conditions un-der which everything is correct. Hint: You use the integrating factor approach. Mul-tiply both sides by eP(t), verify the left side equals d

dt

(eP(t)y(t)

),and then take the

integral,∫ t

a of both sides.

37. Suppose f is a continuous function which is not equal to zero on [0,b]. Show that∫ b

0

f (x)f (x)+ f (b− x)

dx =b2.

Hint: First change the variables to obtain the integral equals∫ b/2

−b/2

f (y+b/2)f (y+b/2)+ f (b/2− y)

dy