Question: I have a sister that is interested in learning *"what I do"*. I'm a 17 years old math loving person, but I don't know how to explain integrals and derivatives with some type of analogies.
I just want to explain it well so that in the future she could remember what I say to her.

Answer: 
Explain derivatives using the speedometer ! Ask her how one could find the speed of something , and if she goes saying the average speed , ask her how one could calculate the instantaneous speed. And boom , there comes your derivatives.
I remember the explanation I was always given.. 

"Do you know how to find the area of a circle" 

"Yes" 

"Well, do you know how to find an area of a random curved figure?" 

"No" 

"That is what we use calculus for".