Show that f is integrable on [0,2] and calculate the integral.

  • Thread starter kingstrick
  • Start date
  • Tags
    Integral
In summary, the function f(x) is defined as 1 for all values of x except for x=1, where it is defined as 0. To show that f is integrable on the interval [0,2], we can use Riemann sums and partitions to show that any sequence of Riemann sums with a maximum interval length approaching 0 will converge to the same value. This can be done by constructing a finer partition that includes the point x=1 and summing the intervals "x<1" and "x>1".
  • #1
kingstrick
108
0

Homework Statement



Let f:[0,2] →ℝ be defined by f(x):= 1 if x ≠ 1 and f(1) :=0. Show that f is integrable on [0,2] and calculate its integral.

Homework Equations





The Attempt at a Solution


i am thinking that the sup{L(p,f)} and inf{U(p,f)} is 1 at every where but where x=1. And I would assume that it is 0 where x=1. So Do I have to Rieman Summations [0,1) and (1,2]. I am confused as to how to approach this when it is not continuous.
 
Physics news on Phys.org
  • #2
Well, just one Riemann sum won't do it- to show a function is integrable, you have to show that any sequence of Riemann sums, with the maximum length of an interval going to 0, converges to the same thing.

If "P" is any partition of [0, 2] we can always make a "finer" partition, P', by adding 1 as a partition point so that L(P', f) is the sum of "x< 1" and "x> 1".
 

Related to Show that f is integrable on [0,2] and calculate the integral.

1. What does it mean for a function to be integrable?

Integrable means that the function can be represented by a specific type of mathematical operation known as integration, which involves finding the area under the curve of the function.

2. How do you determine if a function is integrable on a specific interval?

In order for a function to be integrable on a given interval, it must be continuous and bounded on that interval. This means that there are no breaks or jumps in the function, and the values of the function do not approach infinity.

3. What is the process for showing that a function is integrable on a specific interval?

The process for showing that a function is integrable on a specific interval involves using mathematical techniques such as the Riemann sum or the Fundamental Theorem of Calculus to prove that the function meets the criteria for being integrable.

4. How is the integral of a function on a specific interval calculated?

The integral of a function on a specific interval is calculated using integration techniques such as the definite integral or the indefinite integral. These techniques involve finding the antiderivative of the function and evaluating it at the bounds of the interval.

5. Can you provide an example of showing that a function is integrable and calculating its integral on a specific interval?

Sure, for example, if we have the function f(x) = x^2 on the interval [0,2], we can use the definite integral to show that it is integrable by evaluating the integral of x^2 from 0 to 2, which is equal to (2^3)/3 - (0^3)/3 = 8/3. This shows that f(x) is integrable on [0,2] and its integral is equal to 8/3.

Similar threads

Vector space of functions defined by a condition
2
Replies
58
Views
3K
Centroid calculation using integrals
Replies
9
Views
1K
Expressing a limit as a definite integral
Replies
7
Views
2K
Prove that ∃ a∈]0,2[ such that f(a)=a^2
Replies
12
Views
2K
Prove limit comparison test for Integrals
Replies
2
Views
1K
Integrating an inequality for two functions of the same variable
Replies
9
Views
579
If A is dense in [0,1] and f(x) = 0, x in A, prove ∫fdx = 0.
Replies
2
Views
1K
Double Integral via Appropriate Change of Variables
Replies
3
Views
1K
Complex Integration Along Given Path
Replies
3
Views
1K
How should I use the Jacobi equation to show this?
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top