# Thread: Inequalities & A Definite Integral

1. ## Inequalities & A Definite Integral

I don't understand the first step of the following proof (I was given the first step and I completed it fine, but it is the first step that I do not understand).

First step:

The rest of the proof was straightforward. From the first step, this next part comes from the fact that the resulting integral from the first step is less than or equal to:

Anyways, I just need help understanding why the first step is true. Thanks for any help!!!

2. ## Re: Inequalities & A Definite Integral

Originally Posted by zachd77
I don't understand the first step of the following proof (I was given the first step and I completed it fine, but it is the first step that I do not understand).

First step:

It is a theorem in the theory of integrals:

If $f$ is integrable on $[a,b]$ then $|f|$ is integrable on $[a,b]$ and

$\left| {\int_a^b f } \right| \leqslant \int_a^b {\left| f \right|}~.$

Is that the question?

3. ## Re: Inequalities & A Definite Integral

Yep! That is what I am looking for; I have not seen that property of integrals before.

Thanks for the help!