2: Applications of Integration
- Page ID
- 89245
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In the previous chapter we defined the definite integral, based on its interpretation as the area of a region in the \(xy\)-plane. We also developed a bunch of theory to help us work with integrals. This abstract definition, and the associated theory, turns out to be extremely useful simply because "areas of regions in the \(xy\)-plane" appear in a huge number of different settings, many of which seem superficially not to involve "areas of regions in the \(xy\)-plane". Here are some examples.
- The work involved in moving a particle or in pumping a fluid out of a reservoir. See section 2.1.
- The average value of a function. See section 2.2.
- The center of mass of an object. See section 2.3.
- The time dependence of temperature. See section 2.4.
- Radiocarbon dating. See section 2.4.
Let us start with the first of these examples.
- 2.1: Work
- While computing areas and volumes are nice mathematical applications of integration we can also use integration to compute quantities of importance in physics and statistics. One such quantity is work.
- 2.2: Averages
- Another frequent application of integration is computing averages and other statistical quantities. We will not spend too much time on this topic — that is best left to a proper course in statistics — however, we will demonstrate the application of integration to the problem of computing averages.
- 2.3: Center of Mass and Torque
- If you support a body at its center of mass (in a uniform gravitational field) it balances perfectly. That's the definition of the center of mass of the body.
- 2.4: Separable Differential Equations
- A differential equation is an equation for an unknown function that involves the derivative of the unknown function. Differential equations play a central role in modelling a huge number of different phenomena. Here is a table giving a bunch of named differential equations and what they are used for. It is far from complete.