Integration

Mathematicians have been playing with the concept of integration for ages, but it was until Newton's time that it was realized that integration and differentiation are inverse operations.

In geometry we learned about the concept of area, which is the amount of two-dimensional surface covered by a figure, we also learned how to calculate the area of all kinds of different polygons, using specific formulas.

But the ancient mathematicians realized things get much trickier when curvature is involved.

Say we have a function and it is a curve here, and we want to know the area under this curve over this interval let's say from a to b

That means we are looking at this region S, which is enclosed by the curve, the x-axis vertical lines.

x equals a

x equals b

Then we will realize that there is no formula for this, this is not a rectangle, trapezoid, or any other polygon, as polygons have only straight-line segments for sides.

But the answer has already been told when first learned about differentiation.

We had just learned about how do we get slope and tangent lines through a method of exhaustion. Through that method when we couldn't get the slope of a line with one point, we can make a second point so that we could get the slope, and then pushed the second point towards the first, and then we can get the slope.

same as what limits do.

In fact, In here We can't get the area of this shape so let's approximate it with a shape we can get the area of, a rectangle.

If we divided it into some rectangles and we might see that it's not enough approximation but let's put more of them and make them narrower. so we can just divide it into more rectangles it will be more and more narrow, we are more closely approximating this area. In the limit of an infinite number of infinitely thin rectangles, we will get the precise area under the curve.