Why does integrating the function of a curve give you the area under the curve?

You have probably been taught that if you want to find the area under a function / line, you will integrate the equation of the line. But why is it that when you integrate the equation of the line, you get the area underneath the line? Here is a full explanation. p.s. You will need to know basic limits and how to differentiate using first principles.

[bluewolf]

Great job. Seen 20 videos and this is the most clear!

[electronx]

wow this is the clearest thing ive seen in my 18 years life

[ellen]

Have seen tons of examples but no one explains why it is the way it is. Now I'm convinced. Exactly what I need. Thank you!!!

[Mustafa-cpwc]

I don't understand the limit property he used at 14:00 can someone please send link to explaining it??

[kato]

COMMON QUESTIONS ANSWERED

For those who are confused about the limit part and why that lets the orange area = the pink area.

The idea behind the limit, is to make the approximation accurate. The orange area is approximately = to the pink area because the orange area doesn't account for the extra little bit on top. So if we take a big slice (represented by the variable h), the approximation will be off by quite a bit. But what if make the slice smaller (Smaller h), the approximation becomes better. So then, what if we make the slice infinitesimally small? Then, the pink area becomes the orange rectangle. In other words, the orange area = the pink area.


For those who are confused as to why the integral can sum up all the little pieces of h.

We must remember that the integral F(x), is the formula which gives the area of the original function f(x). A great way one of my teachers put it was, "The derivative is the rate of change of the formula, whereas the Integral is the total change of the formula". The derivative is a function which explains how the graph is changing, whereas the integral is a function which explains how much the graph has changed - the integral F(x) is the total area formula from 0 to x for the function f(x).

As for how definite integrals work, it comes back to the formula A = F(x+h) - F(x) at 4:30. F(x+h) calculates the area from 0 to x+h; and F(x) calculates the area from 0 to x. Then when you take the difference (Just like A = F(x+h) - F(x)), the result is the area between x and x+h.

[howmathematicianscreatemat]

wow, you are honestly the most badass monk I have ever seen^^

[davidagustinus]

Thank you so much, it really good explanation :)

Now i've understood where the formula came from:)

Once again, Thank you so much :)

[SciGuy]

12:58 Great video BUT You didn't show us why the area of the orange bit = the pink bit with your graphing program. That's really the key for me to understand this.

[マウ-xw]

14:11 but h is not a constant, it is what you are doing the limit with

[nolanmullen]

FINALLY! A real explanation!! Thank you, I finally understand :) woo hoo!!!

[studentcommenter]

Thanks amazing explanation. Can you also tell me why you have taken Area to start from 0 to x? i.e. F(x) defines "area under curve from *0 to x*". What would happen if we take different range. Why this specific range only?

[yeetntnt]

so is deviding F(x+h)-F(x) by h at 14:22 the same as differentiating it?

[jameskang]

Hi Magic Monk.

Thank you for the helpful video.
I'm not sure I understand the last bit of the proof where you divide both sides by h.

If we find the limit as h approaches 0 for F(x+h) - F(x) that limit is equal to zero.
In other words, lim F(x+h) - F(x) = 0 = h * f(x) according to the second last line of your proof.
This equation is problematic as the limit should also be applied to the right hand side since if h is approaching zero only for left hand side, the two expressions are not equivalent.

So, the equation should read: lim F(x+h) - F(x) = lim h * f(x).

This means all that the equation has accomplished is just that 0 = 0.
We cannot divide both sides by h because it is part of a limit and the limit of h approaching 0 for h is zero.

Consequently we cannot arrive at your conclusion.

Am I missing something here? Just trying to understand. Your conceptual approach was really helpful but I'm having trouble understanding that last bit.

[AmanKumar-loiz]

Ok but
How it work as summation and add all tiny areas

[eahmeutube]

Please explain Ho Jia Qian’s question here. How is it? Not clear here.

[arbdistress]

Thanks a lot. I understand this is for intuitive reasoning for junior math students helping them to grasp the idea behind and move on to exercise questions. It is great and most students would find it easy to understand. Appreciate the effort and it is important to first let the students get an intuitive understanding instead of being scared by a swarm of greek symbols in the first sight.

However I suggest that we should also point out that the first statement of "proving differentiating area function F(x) = f(x) automatically proves integrating f(x) is the area" not vigorous. For example we cannot handle cases where the function F(x) derivative is not defined. That's why those thick math textbooks almost always starts from Riemann Sum and mesh (i.e. that scary looking proof for young students).

[johnmcgaughey]

The last screen has an error, instead of dy/dx it should be dF/dx, the definition of differentiation is for function F not y=f(x)

[linshi]

thank you.I have watched 5videos and finally understood until this.

[makehimobsessedwithyou]

but it hasn't explain why definite integral instead of indefinite integral

[gireeshbhat]

Ha ha Thank you very much you are the one who give satisfieng reson no any in generel teacher books have this concept