The last subject we covered in class was The Fundamental Theorem of Calculus which brings this course full circle with a proof that proves all the basic things we used in every Calculus class we have been in. I have stated the definition:
Fundamental Theorem of Calculus:
(i) If is integrable and then
(ii) If is integrable, define for x in [a,b]. Then, G is continuous on [a,b]. Furthermore, if g is continuous at a point c in our domain, then exists.
Notice that with some work, we can prove that if we have a continuous function, then the integral of that function is: for some where .
Thus, proving the computation of a proof if the function is continuous has now become 100x.
Note that this proof also allows us to prove many of the properties of integrals.