Many knowledge is connected in math. In my opinion, integration is expressed by anti-derivative form. anti-derivative is the method to do integration. So some students may confused with these two concepts, since they might think they are basically same.

Anti-derivative is the inverse operation of the derivative, which in order to find what value results this result. This is the method via derivative to get the answer.

Integration is the opposite concept with derivative. The integration is to find the area under the certain interval or uncertain interval.

Some students may think they can just use anti-derivative to represent the integration. However anti-derivative does not have interval.

According to these two concepts, we can see that anti-derivative is the method to solve integration problem, since students need find the integration via anti-derivative which can represent the uncertain interval result for integration. So anti-derivative is the crucial way to calculate the  integration.

Math can change the world. The motivation of developing   integer is from the needs of application. In reality, sometimes people can eliminate some unknown variable, but with the development of technique. It makes more and more important to know the exact value. For same simple figures, such as rectangle, circle, triangle… We can use formula to find the area. However, if we need know some complex figures. We have to use integer.

Basically, integer is divide the figure into some parts. That we can refer these some parts as some small rectangles. We can easily find the area of the rectangle (width times length) instead of the complex figure. We just need to add these rectangles together. Then we can get the area of the complex figure.

This method is created by Riemann in 1854. He had a speech to illustrate his method to find the area of complex figure. His discover not only impact the geometry, but also form the fundamental of the Einstein’s relativity. We can literally say, Riemann change the world.

Lebesgue integration solves the problem that Riemann’s integration cannot solve. It moves the integer from 2-d to every dimensions.