Kenneth Kuttler

186 CHAPTER 7. THE INTEGRAL

8. In fact, show there exists a solution to the initial value problem which is to find ysuch that y(t) = y0 +

∫ t0 f (s,y(s))ds under these conditions for t ∈ [0,T ]. Hint: Use

Picard iteration. Let y0 (t) = y0 and if yn (t) has been obtained, let yn+1 (t) = y0 +∫ t0 f (s,yn (s))ds and show, using the Weierstrass M test on a telescoping series that

this sequence converges uniformly to a continuous function y which is the solutionto the integral equation and hence the initial value problem.

9. Give an example of piecewise continuous nonnegative functions fn defined on [0,1]which converge pointwise to 0 but

∫ 10 fn (x)dx = 1 for all n. This will show how uni-

form convergence or something else in addition to pointwise convergence is neededto get a conclusion like that in Theorem 7.2.1.

10. Let F (x) =∫ x3

x2t5+7

t7+87t6+1 dt. Find F ′ (x) .

11. Let F (x) =∫ x

1+t4 dt. Sketch a graph of F and explain why it looks the way it does.

12. Let a,b > 0 and F (x) =∫ ax

a2+t2 dt+∫ a/x

a2+t2 dt. Show that F is a constant. Hint:Use the fundamental theorem of calculus.

13. Here is a function:

f (x) =

 x2 sin(

1x2

)if x ̸= 0

0 if x = 0

Show this function has a derivative at every point of R. Does it make any sense towrite

∫ 10 f ′ (x)dx = f (1)− f (0) = f (1)? Explain. Does this somehow contradict the

fundamental theorem of calculus?

14. ∑nk=1 f (xk−1)(xk − xk−1) ,∑

nk=1 f (xk)(xk − xk−1) are called left and right sums. Also

suppose that all partitions have the property that xk −xk−1 is a constant, (b−a)/n sothe points in the partition are equally spaced, and define the integral to be the numberthese right and left sums get close to as n gets larger and larger. Show that for f givenas 1 on rational numbers and 0 on irrational numbers,

∫ x0 f (t) dt = x if x is rational

and∫ x

0 f (t) dt = 0 if x is irrational. It turns out that the correct answer should alwaysequal zero for that function, regardless of whether x is rational. This illustrates whythis method of defining the integral in terms of left and right sums is terribly flawed.Show that even though this is the case, it makes no difference if f is continuous. Thisintegral was used by Cauchy in the early 1800’s. He considered one sided sums forcontinuous functions and ended up giving the first complete proof of the fundamentaltheorem of calculus.

15. Suppose f is a bounded function on [0,1] and for each ε > 0,∫ 1

εf (x)dx exists. Can

you conclude∫ 1

0 f (x)dx exists? You need to be in the situation of the 1800’s integralto do this problem.

16. Suppose f is a continuous function on [a,b] and∫ b

a f 2 (x)dx = 0. Show that thenf (x) = 0 for all x.

17. Let f be Riemann integrable on [0,1] . Show that x →∫ x

0 f (t)dt is continuous. Hint:It is always assumed that Riemann integrable functions are bounded.

18610.11.12.13.14.15.16.17.CHAPTER 7. THE INTEGRAL. In fact, show there exists a solution to the initial value problem which is to find ysuch that y(t) = yo + Jo f (s,y(s)) ds under these conditions for ¢ € [0,7]. Hint: UsePicard iteration. Let yo (t) = yo and if y, (t) has been obtained, let y,+1 (t) = yo +Jo f (S,¥n (s)) ds and show, using the Weierstrass M test on a telescoping series thatthis sequence converges uniformly to a continuous function y which is the solutionto the integral equation and hence the initial value problem.Give an example of piecewise continuous nonnegative functions f,, defined on (0, 1]which converge pointwise to 0 but fo fn (x) dx = 1 for all n. This will show how uni-form convergence or something else in addition to pointwise convergence is neededto get a conclusion like that in Theorem 7.2.1.3 . .Let F (x) = fo atthe dt. Find F’ (x).Let F (x) = fr a dt. Sketch a graph of F and explain why it looks the way it does.Let a,b > Oand F (x) = fo” ze dt + ae ae dt. Show that F is a constant. Hint:Use the fundamental theorem of calculus.Here is a function:2° sin (2) ifx 40Oifx=0f(x) =Show this function has a derivative at every point of IR. Does it make any sense towrite So f' (x) dx = f (1)—f (0) = f (1)? Explain. Does this somehow contradict thefundamental theorem of calculus?Vie S (x1) (Xe — Xe—1) Ley S (xe) (ae — Xe-1) are called left and right sums. Alsosuppose that all partitions have the property that x; —.x,_1 is a constant, (b—a) /n sothe points in the partition are equally spaced, and define the integral to be the numberthese right and left sums get close to as n gets larger and larger. Show that for f givenas 1 on rational numbers and 0 on irrational numbers, 5 f (¢) dt = x if x is rationaland fj f (t) dt =0 if x is irrational. It turns out that the correct answer should alwaysequal zero for that function, regardless of whether x is rational. This illustrates whythis method of defining the integral in terms of left and right sums is terribly flawed.Show that even though this is the case, it makes no difference if f is continuous. Thisintegral was used by Cauchy in the early 1800’s. He considered one sided sums forcontinuous functions and ended up giving the first complete proof of the fundamentaltheorem of calculus.Suppose f is a bounded function on (0, 1] and for each € > 0, fe f (x) dx exists. Canyou conclude fy f (x) dx exists? You need to be in the situation of the 1800's integralto do this problem.Suppose f is a continuous function on [a,b] and [? f? (x)dx = 0. Show that thenf (x) = 0 for all x.Let f be Riemann integrable on [0, 1]. Show that x > fj f (¢) dt is continuous. Hint:It is always assumed that Riemann integrable functions are bounded.