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
Let’s use an example.
We understand that the derivative of a graph of distance over time gives you velocity over time.
Therefore the anti-derivative of a graph of velocity over time should give you back your distance over time.
So now we’ve transferred the question: why does the area under a graph of velocity over time give you the distance?
Well, easy. Velocity is distance over time. So multiplying that by time gives you back distance as the time and “over time” cancel out (e.g. m/s x s = m).
If you had a graph of constant velocity, then it would be abundantly clear that velocity x time corresponds to the rectangle under the graph, with velocity being the lengths of the vertical sides of that rectangle and time being the length of the horizontal sides.
When the graph is curvy you have to break it up into infinite, infinitesimally small rectangles to get that area, but the area still represents the distance (e.g. the antiderivative) nonetheless.
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