Motivation for Integration

Integration is an abstract mathematical operation of “piling up” to represent a shape. For example, we can imagine the Pyramid from ancient Egypt. As we know, the Pyramid is made of many rectangular solids.  Assuming we can only use the rectangular solids to make a Pyramid, one side of the pyramid is like a stair. In order to make the stair smoother so that the pyramid looks more like the perfect square-based-pyramid, smaller blocks can be used. But if we use smaller blocks, the total size of the pyramid also gets smaller, so we need to use more blocks to keep the rough total size. As we use smaller and more blocks, the steps of the stair will be smaller but the number of the stair will increase. By repeating this process, the pyramid can get closer and closer to the perfect square-based-pyramid. The total volume of the pyramid would be closer to the volume of a perfect square-based-planer as well. Our concept of mathematical integration is analogous to this process. Instead of blocks, we just use rectangles to make a stair along a curve smoother. The size of the pyramid could also influence the final volume of the perfect square-based-pyramid. The higher the pyramid is, the more layers of blocks were used to build the pyramid. When all those blocks for building the pyramid are extremely small, in comparison, the total number of the blocks used in a bigger pyramid is larger than a small pyramid. The difference of the total number of blocks used in two different pyramids with different sizes leads to difference on the size of two square-based-pyramids. This is similar to one of the ways to find the area under different curves.

Ricky Tindjau

Yuze (Chronus) Zhang

Tetsuya (Tim) Matsumoto