How does calculus work?

Most of my experience with calculus involves short-cut equasions to find integrals or derivatives. I understand the concept. For example, an integral is derived conceptually the same way any area is derived--namely by multiplying the height of an intreval by its width, and adding up all of the intrevals. I... show more Most of my experience with calculus involves short-cut equasions to find integrals or derivatives. I understand the concept. For example, an integral is derived conceptually the same way any area is derived--namely by multiplying the height of an intreval by its width, and adding up all of the intrevals. I understand that calculus conceptually does this by making the intrevals infinately small. But HOW does it do this. I don't get how the short-cut equasions result in this? Am I even explaining this right?
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