Antiderivative exists but not integrable In the sense of Riemann, could

Maliyah Robles 2022-07-05 Answered
Antiderivative exists but not integrable
In the sense of Riemann, could an integral (if there exists could you give the less pathological counter-example possible) have an antiderivative but not be integrable on a compact subset?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Kayley Jackson
Answered 2022-07-06 Author has 16 answers
Explanation:
It is possible. Let f ( x ) = x 1.2 sin ( 1 x ) when x ( 0 , 1 ] and f ( 0 ) = 0. It is a differentiable function (check it) in the interval [0,1] and its derivative equals to f ( x ) = 1.2 x 0.2 sin ( 1 x ) 1 x 0.8 cos ( 1 x ) when x 0 and f ( 0 ) = 0. So obviously f′ has an antiderivative in [0,1], but it isn't even bounded, hence not Riemann integrable.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-20
Help with Antiderivative (couple of questions)
I am not very good at this and still trying to understand how it works, but I really need to find a antiderivative of x 6 . Would be very glad if someone could help me with that.
asked 2022-07-08
How to integrate x e x without using antiderivatives or integration by parts.
Yesterday, I sat for my Real Analysis II paper. There I found a question asking to integrate 0 1 x e x d x without using antiderivatives and integrating by parts.
I tried it by choosing a partition P n = ( 0 , 1 n , 2 n , , n 1 n , 1 ) , but I was not able to show that lim n U ( f , P n ) = lim n L ( f , P n ) = 1 .
asked 2022-07-10
Antiderivatives - part c, solving for x
M R ( x ) = 4 x ( x 2 + 26 , 000 ) 2 / 3
I'm already lost at the part 2 u 2 / 3 . How did they get 6 u 1 / 3 + C.
a)Find the revenue function
b) What is the revenue from selling 250 ​gadgets?​
c) How many gadgets must be sold for a revenue of at least ​$50​,000?
Solve for x. (How?)
6 ( x 2 + 26 , 000 ) 1 / 3 150 = 50
asked 2022-06-24
Antiderivatives: Car Deceleration Problem
A car braked with a constant deceleration of 40  ft / s 2 , producing skid marks measuring 160 ft before coming to a stop. How fast was the car travelling when the brakes were first applied?
I know I can solve this problem using kinematics equations from physics; using v f 2 = v i 2 + 2 a d yields an initial velocity of 113 ft/s. However, I am supposed to be using antiderivatives and not physics. So far, I figured that if a ( t ) = 40 then v ( t ) = 40 t + c 1 and d ( t ) = 20 t 2 + c 1 x + c 2 , where c 1 and c 2 are constants. I'm not quite sure what my next step should be... any suggestions?
asked 2022-07-07
Find the antiderivative x = ( 2 / y )
What is the antiderivative of x = 2 y ?
I have to find the antiderivative to do a volume problem, revolving around the y-axis.
I tried doing it, and I think it's 4 y 3 3 π.
asked 2022-05-30
Antiderivative of Antiderivative
Probably easy but I'm not very sure. If f(x) has an antiderivative F(x) then F(x) has also an antiderivative. True or False?
asked 2022-06-16
Physics related antiderivatives problem
I was wondering if I could get a little help with a calculus related word problem:
A car braked with a constant deceleration of 5 meters per second squared for 60 meters before stopping. How fast was the car traveling when the brakes were applied? I can't use the definite integral, only the antiderivative.

New questions