The fundamental theorem of Calculus

Here I am. At the end of my project. My next to last post. And my topic is the fundamental theorem of Calculus. Like the name says, this theorem is a pretty big deal. Let's start by what the Theorem says according to Wolfram Alpha:

I can proudly say I actually understood some of that, but at first could not yet grasp the implications that statement had on all I had studied so far.

Here is Dr. Fowler from Mooculus.com explaining this theorem and it's implications brilliantly:

Now here is what (I think) the fundamental theorem of calculus means in my own words. If you want to integrate acontinuous function on a closed interval, instead of doing the limits of the Riemman Sum applicable, just find the anti-derivative of the function you need to integrate and substract the result of the evaluation of that anti-derivative at the befining and end points of your closed interval.

Or in better words, forget about integrating, just anti-diferentiate!

In my last two posts I had been searching for the area under the curve of x² from the interval 0 to 2. And I had to do a bunch of sigma calculations and set up Riemman Sums and then even take a limit in order to get to 2.667 square units which is 8/3.

The fundamental theorem of calculus is, as I see it, a reward for all my efforts. It is a way of saying:"Fernando, you have toiled and fret, and sweated over these sums and spend countless pages calculating and recalculating all these stuff. You have earned a shortcut." Why thank you very much calculus!

Do you want to check out my new super power? Ok.

The FTOC is telling me that to integrate from the interval a=0 to b=2 of the function x² all I need is an anti-derivative of that formula which I will proceed to evaluate at both points of my interval and then subtract. What is an antiderivative of x²?

Technically there is a + C after that formula but I am assuming it is 0, check Dr. Fowler's video again for that to make sense.

Ok, so now I have my antiderivative x³/₃ and I am ready to evaluate it at 2 (b=2) and at 0 (a=0) and then take the difference .

Now let's plug it back into the formula for the fundamental theorem of calculus.

And finally we get:

There you have it, the elusive 2.667, the area under the curve of x² From the interval starting at 0 ending at 2.

Wow, that took way less effort than before. However, If I had not passed through all those previous steps, all that trouble and effort, I would not have appreciated the beauty, simplicity and deeper meaning of what I have accomplished.

This is my next to last post in this blog and it is a fitting one. I cannot begin to express the emotions I feel at the moment. I have attained great insights in this journey. I now have a deeper understanding of math and the world around me.

Let me know your thoughts in the comments section.