Is it just a coincidence, or is it like an assumption and everything else is built around it?
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I like to think of the reverse process. An integral is the inverse of a derivative. The derivative gives us the slope of the tangent line. The integral gives us the area underneath the curve.
A flat, horizontal line would be described as y=c let’s just say its 5 units up, so y=5. If you want to know the area under this line, whatever region you pick is going to be a rectangle. The height is 5 and the width is the distance along the x-axis. So the integral is 5x.
A slanted line would be described as y=mx+c. Let’s just say the line has a slope of 4 and intersects the origin. The line is y=4x and the area under this curve looks like a triangle where the height is the value for y (we know y=4x so the height is 4x) and the base is the distance along the x axis. So the area of a triangle: 1/2*b*h becomes 1/2*x*4x or 2x²
If the inverse of the integral is the derivative, then we just work backwards to find the derivative. The derivative of 2x² must be 4x. Likewise, the derivative of 5x must be 5. The area of a parabola is not quite so easy but you already know how it’s going to turn out from the trick we use. The easiest way to find out, though, is to split up the function into thinner and thinner rectangles and just add them all up. You’ll see that the answer gets closer and closer to the same pattern.
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