Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := sin √x√1 + x2
Question
Determine whether the following functions are Riemann integrable on [0, 1] or not.(1) f (x) := sin √x√1 + x2
Solution
The function f(x) = sin(√x)√(1 + x^2) is Riemann integrable on [0, 1].
Here's why:
-
The function is defined and continuous on the interval [0, 1]. The square root function, sine function, and polynomial function are all continuous functions. The composition of continuous functions is also continuous. Therefore, f(x) is continuous on [0, 1].
-
The function is bounded on the interval [0, 1]. This is because the sine function always outputs values between -1 and 1, and the square root function and polynomial function always output non-negative values. Therefore, the product of these functions is also bounded on [0, 1].
According to the Riemann integrability criterion, a function is Riemann integrable on an interval [a, b] if and only if it is bounded and continuous on [a, b]. Therefore, f(x) = sin(√x)√(1 + x^2) is Riemann integrable on [0, 1].
Similar Questions
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
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).
Find each of the following integrals:∫ 1√9 − x2 dx
Calculate the following limits:(i) limx→0sin(x2)x + 1 ; (ii) limx→0√x sin(1/x)
Show that the following functions are differentiable at x = 0.(a) |x| sin x (b) x2/3 sin x (c) 3√x(1−cos x) (d) h(x) ={x2 sin(1/x), x 6 = 00, x = 0.Page 3
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.