Hi all :)
Just starting to learn Calculus and we're at limits/derivatives.
Something I always wanted to do but never had the knowledge to is finding areas under curves. People always told me calculus teaches that, but I really don't know how long it'll take until we get there, so I thought I'll use what I currently know and try and find the area under $y=x^2$, between the origin and the point $(4,0)$.
After reading online that calculus uses infinitly thin rectangles and summs their areas, I set to do so myself.
Long story short, I found that if you calculate the area in two ways, one way where the origin is your starting point and another way where the second point $(4,0)$ is your starting point, and then compare both answers, all infinities cancel out and you're left with a simple subtraction where both operands seem to be the derivatives of the original function.
I have two questions, if I may:
The way I did things is by alternating between the two points as the starting point and then comparing the results. Is this the way mathematicians use to calculate the area under curve? I'm sure there is a generalization, but when first figuring things out, is this the way it was done?
I also read online that there is a link between limits\derivative and areas under curves. So far, I could not find the connection between the two. It definitely looks like the derivitives of the function are there, but other than calling it a coincidince, I couldn't find a mathematical way (that is, using algebraic methods) of going from limits\derivatives to the area under a curve. Am I missing something?
Thanks a lot in advance,
Hoping my questions were clear :)
- Arye Segal.