Infinitesimal calculus is the set of tools that allow analysis of continuous variables, that is variables that can take on an uncountably infinite number of values, instead of just discrete variables. Time and space are (for most purposes) continuous, so to accurately analyze any motion more complicated than a constant acceleration, calculus is a necessity. Most applications of calculus in science and engineering involve the behavior of something through time. For example, in classical mechanics, the second derivative of position with respect to time, multiplied by mass, is often set equal to a discovered formula for a force or collection of forces, and then solved for the position. This gives the position of an object for any time greater than zero for any starting position or velocity. This can be used to model many relatively simple systems. A similar application of calculus, and my personal favorite, comes from the calculus of variations and takes the form of Lagrange’s formulation of classical mechanics. His equations can help analyze systems that may be very complicated with Newtonian mechanics. The importance of being able to study the motion of objects through time when the forces are known is obviously important, but calculus does not necessarily deal with time. Calculus can be used to analyze volumes, areas, and centers of mass of complicated shapes, and how those may change. From geometry, we know how to find the volume of a cylinder or a cone, but we need calculus to find the volume of general surfaces of revolution. There are a lot of times, however, when calculus provides a solution to a problem that is very complex and actually cannot be expressed in the typical functions used in mathematics. This may seem like an issue, but I think that it just illustrates the world is more complicated than what we can currently completely understand. Also, calculus, as well as many other areas of mathematics such as linear algebra, provides us with ways of getting really close to the answer, using certain methods of approximation. An example of this would be a Taylor approximation, which can turn an unfriendly function into a nice polynomial of any degree depending on the desired precision, and it also allows us to know the biggest possible error that could occur with the method with the Lagrange Error Bound. These are just some of the many, many applications of calculus because it is so prominent in any discipline that uses math. I think it would be extremely hard to find an area of science or engineering or technology which has not been touched by the study of infinitesimals.