14.8.3: Applications of Integration
- Page ID
- 67226
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In this section, we use definite integrals to calculate the force exerted on the dam when the reservoir is full and we examine how changing water levels affect that force. Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world around us.
- 14.8.3.1: Prelude to Applications of Integration
- The Hoover Dam is an engineering marvel. When Lake Mead, the reservoir behind the dam, is full, the dam withstands a great deal of force. However, water levels in the lake vary considerably as a result of droughts and varying water demands.
- 14.8.3.2: Volumes of Revolution - Cylindrical Shells
- In this section, we examine the method of cylindrical shells, an example for finding the volume of a solid of revolution.
- 14.8.3.3: Arc Length of a Curve and Surface Area
- The arc length of a curve can be calculated using a definite integral. The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of y. The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution. The integrals generated by both the arc length and surface area formulas are often difficult to evaluate.
- 14.8.3.4: Physical Applications of Integration
- In this section, we examine some physical applications of integration. Several physical applications of the definite integral are common in engineering and physics. Definite integrals can be used to determine the mass of an object if its density function is known. Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.
- 14.8.3.5: Moments and Centers of Mass
- In this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have seen performers who spin plates on the ends of sticks. The performers try to keep several of them spinning without allowing any of them to drop. Mathematically, that sweet spot is called the center of mass of the plate.