Why is it that when you integrate a function, you get the area under the curve?
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@experimentX @ParthKohli @mayankdevnani
The area between the graph of y = f(x) and the x-axis . This formula gives a positive result for a graph above the x-axis, and a negative result
If the graph of y = f(x) is partly above and partly below the x-axis, the formula given below generates the net area. That is, the area above the axis minus the area below the axis.
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"Why is it that when you integrate a function"
This is a little bit broad (there are functions that u can integrate that do not give the area under "the" curve).
Maybe you mean over an interval....
Area has a unit of meters squared, right? (m^2) So in order to do that, we have a function with a bunch of lengths.
Since there are an infinite amount of points on a graph, we do magical calculus trickery and multiply each point on the graph by it's infinitesimally small width that's nearly 0 but isn't.
Then we add them all up and get the area under the curve.
So for example, if we have the graph y=2x and want to know the area under the curve from 1 to 3 we would set up this:
Give yourself a second to really admire what's going on here. The big "S" is literally the letter S and it means "sum" all the areas from 1 to 3. And inside we see that we have 2x which is a height and dx which is a width, giving you an area.
Now if we were to do this by hand (which is impossible) we would just start saying:
(2*1)*dx+(2*1.00000001)*dx+(2*1.00000002)*dx+(2*1.00000003)*dx+... until we got to 3. Of course, I showed the decimals as ending at around 7 decimal spaces, but really that's just to convey the idea across since there are really points between the ones I'm adding up there.
So magically we add up an infinite number of rectangles with infinitely small area and somehow it's becomes a real area, pretty weird.
Another example to show that it's an area, let's consider the square here and find the area with both geometry and calculus!
in fact you do not
you get the area under the curve if the function is positive
it is an fact the definition of the area, the limiting value of sum of the area of the rectangles as the bases gets smaller and smaller