Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := sin √x√1 + x2 ,(2) g(x) :=(x sin(1/x) if x > 0,3 if x = 0.(3) h(x) := 1n if 1n + 1 < x ≤ 1n for n ∈ N
Question
Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := sin √x√1 + x2 ,(2) g(x) :=(x sin(1/x) if x > 0,3 if x = 0.(3) h(x) := 1n if 1n + 1 < x ≤ 1n for n ∈ N
Solution
(1) The function f(x) = sin(√x)√(1 + x^2) is Riemann integrable on [0, 1]. This is because it is a continuous function on a closed interval [0, 1], and all continuous functions on a closed interval are Riemann integrable.
(2) The function g(x) is not Riemann integrable on [0, 1]. This is because it is not defined at x = 0. However, if we define g(0) = 0, then g(x) becomes Riemann integrable on [0, 1]. This is because g(x) is continuous on (0, 1] and has a removable discontinuity at x = 0.
(3) The function h(x) is not Riemann integrable on [0, 1]. This is because it is not defined for all x in [0, 1]. However, if we define h(x) = 0 for x not in the form 1/n, then h(x) becomes Riemann integrable on [0, 1]. This is because h(x) is a step function, and all step functions are Riemann integrable.
Similar Questions
Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := sin √x√1 + x2
You Let f : [0, 1] → R, where f(x) := (1 if x ∈ {1/n: n ∈ N},0 otherwiseNote that f is Riemann integrable on [0, 1] by a previous assignment. Let F : [0, 1] → R, F(x) := integral 0 to x (f).(a) Prove that F is differentiable on [0, 1].(b) Find a c ∈ [0, 1] such that f is discontinuous at c and F'(c) = f(c).(c) Find a d ∈ [0, 1] such that f is discontinuous at d and F'(d) 6= f(d).Remark: This shows that if the indefinite integral F of a Riemann integrable function f isdifferentiable at a point c where f is discontinuous, then F'(c) may or may not equal f(c).
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g(x) :=(x sin(1/x) if x > 0,3 if x = 0
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