I struggle with integration, and don't understand why we need to do it

First, my approach is to ask student to count the squares on the area under a curve. This process is tedious, and only gives an estimation. Next, I show them the formula to do integration, and immediately they can appreciate that integration is simply just a mathematical method of finding the area under the curve. increase the power by 1, divide by the new power Why is this method useful? (advance discussion to spark interest for more able students - more relevant to students considering and wants to discuss university application) Many profiles can be defined as mathematical curves, for example how much water flow in a pipe, number of cars past through a location at a time period. Without having to calculate the cars one by one, we can just 'integrate', and it adds the total number of cars. This is useful for example if the local planning wants to know whether to repair the road, or add more traffic signals. So Maths is used behind decision making to provide supporting evidence. And being able to do maths gives you the chance to do exciting stuff!