The Story of Integration (3 of 4): The Relation to Derivatives

[swalehasaleem]

This guy has so much tolerance. Your videos are great. Love from Pakistan!

[cameronboone]

Great video Eddie. I'm a practicing electrical engineer (~14 years post college now) that came across your channel while doing some math review. I was really good at calculus in high school and college but don't have to do calculus very often for my job except maybe a first order derivative for capacitor or inductor calculations every now and then. As I was going back through my math background I realized at times that there was a small hole or gap in my thinking, which can be frustrating when you want to take something back to first principles. It's nice to be able to do calculus but I prefer to understand what I'm doing, if possible. You almost lost me towards the end when you mentioned how you only need the starting and ending points with the primitive function, although I know it is true in terms of how to DO calculus, but I wanted to understand the why and also as you said how gradient (as I'm sure you know, we call it slope in the states :- )) relates backwards to "Area under a curve". I had my "aha!" moment when I realized that what we are really integrating and adding are tiny slopes/gradients times the "run". So when you look at the function you are integrating, the value of that function at each value of x is the slope of the primitive function at that value of x. So when you see that, then integration dealing with the areas makes perfect sense. Only using the starting and ending values made sense when I saw that the rise component for slopes for each successive tiny area are y1 - y0, y2-y1, y3-y2 etc.... So when you add all of those up, they all cancel out except for the first and last. That brought to my mind something I'm familiar with in engineering which is the concept of conservative fields, for example the electric field. When moving a charge through an electric field and calculating the change in electric potential, it does not matter what path the charge takes. Only the starting and end points matter, which makes sense because potential is the integral of Electric Field along the path of travel. Thanks again Eddie for your insightful videos! It is truly great when those "aha!" moments come, even if it's 14-15 years late :-D

[matthewwhite]

Completely spectacular! Beautiful and intuitive example!

I find it not at all weird that integration is the opposite of differentiation. If you move the right border of an area to the right, the area grows by the width times the average height of the newly added area. So you just need the original function again to know how much the area grows. And how much something grows is the derivative, it's how much it increases (over some distance).

[jennawalker]

Bless you, I thought I was going to fail my midterm tonight, I now have hope because I'm actually beginning to understand it. thank you.

[fatimahalmeer]

Absolutely the best video I have ever come across! Thank you so much

[filipcacic]

Respect for clear explanation and enthusiasm

[missghani]

love your explanations Sir, thank you once again.

[gautamsahni]

Thank you thank you thank you thank you...

[sauravthegreat]

Hello, i just had a simple question.
Can it make intuitive sense as to why dy is equal to the area under the curve. Mathematically, you did prove it but intuitively im still kinda confused. I also feel even more confused as dy is one dimensional while area is two dimensional.

Help would be appreciated thanks

[cursebr]

Difference in F(b)-F(a) it’s just average speed multiply by distance which is the area

[abiyermias]

Speed times time would have a unit of distance, not an area?

[Brian-dnny]

whoever yelled "it's all clear now" at 5:45 had me dying

[fireemblem]

To all my American viewers: Gradient = Slope