Why Are Slope and Area Opposite: The Fundamental Theorem of Calculus

[whatitmeans]

3Blue1Brown has a beautiful video about this topic, but your explanation is much more intuitive. Congrats.

[danieldonnell]

They should actually just print this explanation on the cover of calc 1 textbooks... Very concise and clear. Thanks!

[hi-mveg]

This was a brillant explanation never really understood what slope had to do with calculus thanks for clearing it up.

[themonrovian]

After doing an entire physics degree I never saw an explanation as clear as this for illustrating the fundamental theorem of calculus. Bravo sir 👏

[frendlyleaf]

Thanks I wanted to know this exact thing!

[SATMathReview]

Brilliant, amazing we live in an era where such profound things that took so much genius and effort to discover are readily accessible to anyone.

[Yues]

Real thanks bro, you can explain it in really simple term, i dont know is that simple, real thanks!

[sadornsamdi]

I have bien looking for such a video for quite à long time thank you

[mmgedi]

Wow, as the width of the rectangle tends to zero, the height of the rectangle tends to the slope of the original function. In other words, that infinitesimal sliver becomes the tangent line at that specific instant of time. Thank you so much!

[JavierBonillaC]

That is really very good.Every single rectangle has as a height the original function and as width dx. So if the slope is constant (say a horizontal line) the area will alyas be dx times 1. I seem to understand. thank you so much for this super interesting video.

[bismajoyosumarto]

Whoaaa thanks! :D That was quite short and clear 👍

[benbearse]

The way that let me understand integrals the best is that the anti derivative of a function is literally the area formula for under the graph. Like take y=x for example. The distance between any point is X and the height of any point is Y which equals X. This forms a triangle because it’s just a straight diagonal line. The formula for the area of a triangle is bh/2 so base (x) times height(x) divided by 2 =(x^2)/2
Which I thought was really cool. This continues for every other possible line
The area for under a quadratic is 1/3(bh) where base is still x and height is x^2 (hence y=x^2) so (x^3)/3 is the area

[umachakraborty]

I totally understand calculus.☺️☺️☺️thanks.

[myonn]

This should be in every calculus textbook!

[gravitystorm]

Loved the video, great explanation! I have a question though; when you divide by dx and then take the limit, on the right side of the equation you’d have something of the form 0/0, does this matter?

[binodtharu]

Thanks a lot for ur explanation i am very happy right now that i was able to prove this result because of you liked and subscribed👍

[hoteny]

Thanks. Even tho I am free from calc-1 now, it was a good watch.

[BilalAhmed-onkd]

I have a q about this, we must've added the limit as dx->0 before the last step

So we have

Lim dx->0[g(x+h)-g(x)]=f(x)dx
So when we divide by dx we have

{ Lim dx->0[g(x+h)-g(x)] }/dx=f(x)
So what we actually have now is that the numerator applies only on the numerator of the lhs, which is not exactly what the derivative of a function is

[Yguy]

I don't understand why such an important explanation was not in my book. Nice explanation though, Keep it up!

[roygreen]

I think that great explaining ; but ill like to know, why in the first place you can call the area under the curve a function g(x)?