## FIRST FUNDAMENTAL THEOREM OF CALCULUS

For now, we have studied the definition of definite and indefinite integration. However, it would be a labor to compute Riemann sums every time we approach an integration problem.

Luckily for us, Newton and Leibniz, the Brit and the European (logical error here), discovered the relationship between integrals and derivatives 300 years ago.

There finding can be summarized with the First Fundamental Theorem of Calculus. Though this is not as useful as the second one we are about to study, it serves as a step stone to the second.

Luckily for us, Newton and Leibniz, the Brit and the European (logical error here), discovered the relationship between integrals and derivatives 300 years ago.

There finding can be summarized with the First Fundamental Theorem of Calculus. Though this is not as useful as the second one we are about to study, it serves as a step stone to the second.

If F(x) satisfies this, then:

## A Way to Think About This

If you have prior experience in statistics, this is going to be much easier for you.

Think of F(x) as a gigantic paintbrush. It is brushing the area beneath the curve of f(t). So, the paintbrush starts at a value a, and at F(x), paints to x. It is only logical to assume that the speed at which F(x) increases (it's derivative), is equal to the next area it's going to brush. Now, as in calculus, the idea of infinity kicks in here. What if the next area is arbitrarily thin? It's just the value of the function evaluated at t, the next x in line as the paintbrush moves along the x axis.

Now, in statistics, you might call F(x) the cumulative density function, the CDF, while f(x) is the probability density function, the PDF. Makes sense now? Statisticians?

Think of F(x) as a gigantic paintbrush. It is brushing the area beneath the curve of f(t). So, the paintbrush starts at a value a, and at F(x), paints to x. It is only logical to assume that the speed at which F(x) increases (it's derivative), is equal to the next area it's going to brush. Now, as in calculus, the idea of infinity kicks in here. What if the next area is arbitrarily thin? It's just the value of the function evaluated at t, the next x in line as the paintbrush moves along the x axis.

Now, in statistics, you might call F(x) the cumulative density function, the CDF, while f(x) is the probability density function, the PDF. Makes sense now? Statisticians?

## Drawbacks

After studying the first fundamental theorem of calculus, we have one question. The question is-- how the hell do we apply it on all definite integrals that passes through our hands? The answer is-- we can't. The reason is quite simple-- you should be able to tell why just by looking at the equation: this shit is for those "concept" multiple choices. In a test, they might dig out f(x) and give you F(x) and ask for their relationship. Beyond that, the first fundamental theorem of calculus carries less weight in the real world than a chicken turned upside down. (Save for its bridge to the second fundamental theorem of calculus, which is the KICKER!)