Why is the result of the integral the entire area under the function?
In: 2
You can think of an integral as having two important pieces.
1) On it’s own it’s an infinitely thin slice of the area below a function, a magnitude really. A value with no side-to-side measurement (because it’s infinitely thin) and purely a vertical measurement, a ‘height’.
2) The assumption is you are defining a starting point and an ending point for the integration. So you are *integrating* (adding together) a infinite number of infinitely thin slices from the start to the finish. Now the widths *do* add up to something measurable *and* you have all the heights already. In a sense you now have a shape and you can width x height to get the area of the shape. That’s essentially what you are actually doing. Instead of calculating a base x height area for an easy rectangle, you are calculating a complicated base x height for a wonky shape.
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