1. Riemann Integration

This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help

2. I believe this what they call a improper integral, and because it is obviously undefined at 0, for something like this I believe you will have to do

$\displaystyle \lim_{x\to0+}{\int_B}^1\frac{1}{x}$

This should get you started

3. What is B in the integral you entered? I don't understand that.

4. Originally Posted by xyz

This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
Note that over any subinterval $\displaystyle [a,b]\subset [0,1]$ that $\displaystyle \inf_{x\in[a,b]}f(x)=0$....so

5. Is $\displaystyle f(x)=0$ everywhere with the exception of $\displaystyle x=\frac{1}{n}$ , $\displaystyle n \in \mathbb{N}$ so that you can devide the integration interval in subintervals of the type...

$\displaystyle \frac{3}{4}<x\le 1$ , $\displaystyle \frac{5}{12} < x \le \frac{3}{4}$, $\displaystyle \dots$ , $\displaystyle \frac{n+\frac{3}{2}}{n+2}< x \le \frac{n+\frac{1}{2}}{n+1}$ , $\displaystyle \dots$

... so that in each interval is $\displaystyle f(x)=0$ with the only exception of $\displaystyle x=\frac{1}{n}$, i.e. in only one point. In each interval the function is Riemann integrable and is...

$\displaystyle \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (1)

From (1)...

$\displaystyle \int_{0}^{1} f(x)\cdot dx = \sum_{n} \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. Thanks a lot for all the help. I already got the idea from Drexel's hint. It was easy after that. But your solution is also very interesting!
The text passage contains a discussion about solving a specific integral, with multiple users providing input and explanations. One user poses the initial question, and others provide hints and a detailed solution.
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1. Riemann Integration

This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help

2. I believe this what they call a improper integral, and because it is obviously undefined at 0, for something like this I believe you will have to do

$\displaystyle \lim_{x\to0+}{\int_B}^1\frac{1}{x}$

This should get you started

3. What is B in the integral you entered? I don't understand that.

4. Originally Posted by xyz

This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
Note that over any subinterval $\displaystyle [a,b]\subset [0,1]$ that $\displaystyle \inf_{x\in[a,b]}f(x)=0$....so

5. Is $\displaystyle f(x)=0$ everywhere with the exception of $\displaystyle x=\frac{1}{n}$ , $\displaystyle n \in \mathbb{N}$ so that you can devide the integration interval in subintervals of the type...

$\displaystyle \frac{3}{4}<x\le 1$ , $\displaystyle \frac{5}{12} < x \le \frac{3}{4}$, $\displaystyle \dots$ , $\displaystyle \frac{n+\frac{3}{2}}{n+2}< x \le \frac{n+\frac{1}{2}}{n+1}$ , $\displaystyle \dots$

... so that in each interval is $\displaystyle f(x)=0$ with the only exception of $\displaystyle x=\frac{1}{n}$, i.e. in only one point. In each interval the function is Riemann integrable and is...

$\displaystyle \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (1)

From (1)...

$\displaystyle \int_{0}^{1} f(x)\cdot dx = \sum_{n} \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. Thanks a lot for all the help. I already got the idea from Drexel's hint. It was easy after that. But your solution is also very interesting!
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The text passage contains a discussion about solving a specific integral, with multiple users providing input and explanations. One user poses the initial question, and others provide hints and a detailed solution.

TYPE: SINGLE_QUESTION_WITH_ANSWER