Suppose that is integrable on [a,b].
Suppose that is integrable on [a,b]. → R is positive and integrable. Show that, f f(x)...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except (6)...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except (6)...
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
4a. (5 pts) Let f, g: [a, b -R be integrable. Show that la, blR, {f (x),g x)) h h (x) max and k[a, bR, k (x) min {f (x),g (x)) integrable. Hint: Observe that, for all a, b e R, max{a, b}= (a+ b+ la - bl) and min{a, b} (a+b-la -bl). are
8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may not use the theorem that monotone functions are integrable! 8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may not use the theorem that monotone functions are integrable!
Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length 2π Hint: I is contained in two consecutive intervals of the form (kT, (k+2)π) Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
[24] Suppose f: [a, b] -[0, 00) is Riemann integrable with respect to a. Show fp is also Riemann integrable with respect to a over [a, b] for any p> 0. [24] Suppose f: [a, b] -[0, 00) is Riemann integrable with respect to a. Show fp is also Riemann integrable with respect to a over [a, b] for any p> 0.
4. If f is continuous on R and integrable and F(x)-f(t)dt, show that f(x) -70 2e
20.1. Show that if f : [a, b R is of bounded variation, then it is integrable on [a, b]. ; (bi: i)(DULO(ld rl. VZLilin.l.ld DILL. 20.1. Show that if f : [a, b R is of bounded variation, then it is integrable on [a, b]. ; (bi: i)(DULO(ld rl. VZLilin.l.ld DILL.