So I get the idea that the area underneath a part of a graph is the same as the sun of all the infinitely small rectangles that make it up. I also get that you can find the area by between two points by plugging in two values into the equation for the anti differential and finding the difference. I just don’t understand why. What has the anti differential even got to do with anything? Where did it come from and how does it relate to the rectangles? I’ve tried finding other explanations online and on elsewhere on Reddit but it still doesn’t make much sense to me.
Edit: I meant anti derivative not anti differential lol
In: 1
The connection comes from the definition of derivative…it’s usually just referred to as “the slope”, but it’s really, “If I take successively tinier width slices, what value does the average slope converge to?”. Average slope is “height at one edge minus height at the other edge divided by width.” And the area of *that* shape (a tall thin “rectangle” with a sloped top) has *exactly* the same area as an actual rectangle who’s height is the average of the left and right sides (you can doodle on paper to visualize it).
So adding up a bunch of tiny rectangles whose height is the curve is really the same as adding up a bunch of tiny “sloped top slices” that hug the actual curve. And the integral (anti-derivative) is, mathematically, the area of adding up all those little sloped-top slices.
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