Integrating fractional part function

Question

If $f(x) = \min(\{x\},\{-x\})$, then find $$\displaystyle\int_{-100}^{100} f(x)\, {\rm d}x,$$ where $\{x\}$ denotes fractional part of $x$.

mniip · Accepted Answer · 2015-07-12T15:49:05.463

3

Geometrically, the plot of $f(x)$ on $[0, 1]$ is an isosceles right triangle with a hypothenuse (provided by the $x$ axis) equal to $1$. The area of such triangle is $1/4$.

$f(x)$ is periodic, and its period is $1$, so on $[-100, 100]$ the plot is just that triangle repeated 200 times. The cumulative area of those triangles is $200\cdot1/4 = 50$ which is the value of the definite integral.

edited Jul 12 '15 at 15:49

answered Jul 12 '15 at 15:46

mniip

450

The answer my book states is 50... Can you please reconfirm whether it is 100 or 50? – Kushashwa Ravi Shrimali Jul 12 '15 at 15:48
@KushashwaRaviShrimali Oops my bad, arithmetic mistake. – mniip Jul 12 '15 at 15:49
Can you please explain how the triangle is isosceles? – Kushashwa Ravi Shrimali Jul 12 '15 at 15:54
1

$f(1-x)=min({1-x},{x-1})=min({-x},{x})=f(x)$. The graph is symmetric about the line $x=\frac{1}{2}$ – mniip Jul 12 '15 at 15:56
I am unclear that how did you get the hypotenuse as 1? I thought base will be 1 as it is on [0,1] and for this interval f(x) will lie between 0 to 1 also. So, base and perpendicular should equal 1 and hypotenuse should equal sqrt(2) accordingly. Where am I going wrong? – Kushashwa Ravi Shrimali Jul 12 '15 at 15:59
Also, on [0,1] f(x) = x ... So, integral of f(x) from 0 to 1 will be 1/2 ... ? I know I am wrong somewhere but can you please help me realize the mistake? – Kushashwa Ravi Shrimali Jul 12 '15 at 16:03
1

@KushashwaRaviShrimali $f(0.7)=min({0.7},{-0.7})=min(0.7,0.3)=0.3$ – mniip Jul 12 '15 at 16:12
Woah, thanks a lot for patience and answers. – Kushashwa Ravi Shrimali Jul 12 '15 at 16:14

score 2 · Answer 2 · answered Jul 12 '15 at 16:13

2

as mniip said,there are almost 200 triangle in the [-100,100](as you can see from the graphic of function),square of one triangle equal to 1/4,so the answer will be 200*1/4=50 enter image description here

answered Jul 12 '15 at 16:13

haqnatural

21,578

This answer made the graphical overview more clear to me. Thanks a lot.. – Kushashwa Ravi Shrimali Jul 12 '15 at 16:15
1

you're welcome) – haqnatural Jul 12 '15 at 16:20

score 2 · Answer 3 · edited Jun 11 '22 at 08:09

2

$$ x=n+p\\x=\left \lfloor x \right \rfloor +\begin{Bmatrix} x \end{Bmatrix}$$ so we can write $\begin{Bmatrix} x \end{Bmatrix}=x-\left \lfloor x \right \rfloor $

now note that f(x) is even function $$f(-x)=min(\begin{Bmatrix} x \end{Bmatrix}),\begin{Bmatrix} -x \end{Bmatrix})=f(x)$$so $\int_{-100}^{100}f(x)=2\int_{0}^{100}dx=2*100\int_{0}^{1}f(x)dx$ $\space $It is because f(x) is periodic function
enter image description here

$$\int_{-100}^{100}f(x) dx=200\int_{0}^{1} dx=200\cdot \frac{1}{4}=\boxed{50}$$ implicit ,it is a continuous function

edited Jun 11 '22 at 08:09

Myst1cal

267

answered Jul 12 '15 at 16:28

Khosrotash

24,922

Consider adding a \cdot because otherwise it looks like $200+\frac{1}{4}$ – mniip Jul 12 '15 at 16:45

Integrating fractional part function

3 Answers3