Hello fello mathematicians,
I have been learning calculus recently for the first time, independently, and I have hit a wall. In studying the definite integral in depth, I have failed to see how it calculates the area under a curve. I understand that the area under a given curve bounded by two points is the sum of all rectangles as their width approaches zero. I can see that as a sum which is equal to the area under a curve. But I dont see how takes the difference between the anti-derivative at 2 points is equivalent to the area under the curve. I have read two different textbooks and how they approach it, and even online articles, and I understand everything but this relation. May someone please help or provide a better proof or intuitive explanation. The difference is two anti-derivatives of a function to me seems like just taking the difference of two infinitesimally small areas(rectangles). Wouldnt we have to add up all those infinitesimally small rectangles bounded by two points to find the area? I know that is true, but how is that the same as taking the difference between two anti-derivatives at points b and a?